As explained in the second quote (and those after it) — through them:

"And his entire intention in his service and prayer is that the service and prayer should reach the righteous one, and he will raise it to the place where it needs to go"

If so, then at least according to Maimonides this looks like idolatry (worship through intermediaries). And it requires further examination whether that is necessary.

Why is prayer to a being other than the Holy One, blessed be He, considered idolatry? Prayer comes from the sense of judging, hoping, yearning. Is it forbidden to hope for or plead for things from flesh-and-blood people? Beyond that, prayer through a righteous person can be considered like speaking through a telephone—the telephone is just a means of reaching the one I really want to speak to.

If someone wishes another person good health, fine. But to pray in the religious sense is forbidden.
For prayer through intermediaries, see Maimonides at the beginning of the laws of idolatry.

Rashi disagrees with Maimonides and holds that it is permitted, and perhaps even preferable, to pray through intermediaries (angels and the like). I don't have the books in front of me, but I think it's in Sotah.

And Copenhagen's quote is not clear whether this is a later interpretation or something Rabbi Natan himself wrote. In any case, the connection sounds like a kind of hitchhiking, like what the Beresheet spacecraft did.

Or to put it simply: from the fact that people commonly say the liturgical poems "Support my gratitude through the Thirteen Attributes" and "Those who usher in mercy," it seems that Jewish law was not ruled in accordance with Maimonides, and it is not clear what the Rabbi wants from the Breslovers.

There is no halakhic ruling here. Whoever is concerned is concerned, and whoever isn't, isn't. By the same token you could say that since the Breslovers and Hasidim generally accept connecting oneself to the righteous person, that itself proves they do not rule like Maimonides. And indeed I noted that I do not know whether his words are necessary.

I don't understand what halakhic ruling has to do with this. First of all, there is some meaning to what counts as idolatry, and only afterward comes halakhic ruling. It's similar to what Rabbi Shimon Shkop says: first there are basic definitions of what counts as ownership, and only afterward does the Torah come and forbid theft and robbery; it's not that the prohibitions of theft and robbery define what ownership is.

In other words, you can't define the essence of idolatry according to halakhic ruling, because it's a matter of reality—are you serving the Holy One, blessed be He, or something else (some created being)? Once your theory says that a certain act is idolatry, then even if all the sages of Israel permit it—the matter would still be forbidden.

That isn't precise. The definition of what idolatry is not purely factual. It has evaluative components. For example, there are opinions among halakhic decisors that Noahides are not warned against associationism. That doesn't mean associationism is not mistaken on the factual plane. But the prohibition is defined within the framework of Jewish law. All the more so in our case of prayer through intermediaries, which is not about facts at all but only about prohibition and permission.

This too is a factual question: did the Holy One, blessed be He, forbid you to direct the prayer as far as the righteous person so that he would supposedly see to sending it on to Him? The prohibition that the Holy One imposed (or not—according to those who deny it) is a fact of reality.

In my humble opinion, the prohibition is learned in two ways: first, from universal rational morality, which recognizes that one must not connect to created beings with the spiritual and emotional attachment fitting for their Creator; and by contrast, when one sees to raising the prayer only as far as the righteous person, what takes place is a substitution of the object of attachment to a person or some dead soul rather than to God. Second, from the knowledge that the words of the prophets are true, and one who studies the plain meaning of their words with intellectual and spiritual honesty can see that this is abhorrent in His eyes.

By that logic every halakhic question is factual, and then there is no place or meaning for disputes. This is empty semantic pilpul.
As for the rest, these are reasonings, and that is exactly what the disputes are about. But beyond that, a prohibition is grounded in command, not in reasoning. Reasoning can teach that an act is good or bad. Prohibition and permission are matters for Jewish law.
As for "why do I need a verse? It's a logical inference," see my article here, "The Halakhic Status of Logical Reasoning."

Seemingly, the question is whom the righteous person represents.
If one turns to him as a representative of the Holy One, blessed be He, then according to Maimonides this is certainly idolatry.
But if he is the representative of the one praying, then maybe not, since no one maintains that a person who asks his friend to pray for him is among idol worshipers (even if it isn't idolatry, that doesn't mean it's desirable).

Indeed. There is also the further problem that the intermediary is a dead person.

To the Rabbi,

What I argued does not imply that every halakhic question is factual. When you know the fact, no halakhic question arises. Suppose you know that a certain meat is definitely milk or definitely leavened food (on Passover)—there is no halakhic question, only the fact that the Holy One forbids it to you. A question arises when the matter is not known, or when the Holy One left the matter to the intellect of certain people in a state of doubt. Then it may be argued that perhaps there is no fact (or alternatively that the fact is that you are obligated to follow the ruling that emerges from the halakhic discussion).

What is the problem with this being reasoning? The claim that Moses received the Torah from Sinai also rests on reasoning (historical or otherwise). That does not make the opposing opinion equally legitimate in God's eyes.
And after all, the claim that since these are reasonings, therefore the permissive opinions supposedly have some serious standing, is itself also a reasoning.
Prohibition and permission, as I understand them, are what the Holy One forbids or permits (and since the Holy One and His wisdom are one, the moral good and evil that a person reaches through natural reason are identical with the divine command).

I can't manage to follow the discussion here. You wrote that this is a factual question whether the Holy One forbade it or not. From that, the conclusion I wrote indeed follows.

Reasoning is certainly not invalid, but reasoning is not a fact (contrary to what you wrote). That's what I wrote at the beginning. After that I added that there is disagreement about reasonings, and Jewish law does not emerge from reasoning but from command. Reasoning can interpret a command but not generate it (at the level of Torah law). Your last sentence reflects a misunderstanding of the nature of Jewish law. I referred you to my article on reasoning; see there.

An explanation was written regarding connecting prayer to the righteous person—its meaning and how it works—with proofs from major kabbalists, major Hasidic figures, and also Lithuanian rabbis. Worth reading once and for all.

Connecting the prayer to the righteous person means that before prayer one should intend that he is praying according to the mind of the righteous person, and during prayer he should intend that he is praying in connection and inclusion with him, so that the righteous person will complete and repair the prayer. But the intention is not to pray to the righteous person himself, or to think that he raises his prayer through the soul of the righteous person and asks him to raise the prayer; rather, that he should pray with us, like communal prayer, and when the righteous person prays he raises all the prayers.

Sources for the idea of connecting prayer to the righteous person and its acceptance:
Rabbi Jonathan Eybeschutz:

"Thus all prayers revolve around the prayers of the righteous, and attach themselves to their prayer, and he [the righteous person] is one for whom no barrier separates between him and God; through his prayer the prayer of others also rises… and through their prayer the prayer of others also rises."

— Ya'arot Devash, Part I, end of Derush 9
Rabbi Chaim Yosef David Azulai:

"When there is one righteous person in the generation, the Divine Presence is with him, and through him it receives the praise and prayers of the worthy of Israel and their service, by means of that righteous person, the standard-bearer."

— Midbar Kedemot 90:6

Rabbi Moshe Cordovero:

"And their prayer and service are by means of that worthy one, the standard-bearer… and all the prayers of Israel ascend by way of this ladder."

— Elimah, Ein Kol Tamar Heh, chapter 14
The Chatam Sofer:

"And when they [Israel] cleave to one another, then even though I myself cannot direct my prayer properly, I pray according to the intention of the greatest one of the generation; and second, that we connect to one another and all become as one… and if so, we agree with our great ones and make them agents in our mission."

— Chatam Sofer on the Torah, parashat Shemot
The Ben Ish Chai:

"For usually a person's prayer is not complete in terms of intention, but its lack is completed by its joining with the prayers of the righteous, which are complete, and they draw along the prayers of the weak, like a steamship pulling along a simple ship. Therefore one who has no desire and will for connection and unity—measure for measure, his prayer will not be joined with the prayers of the complete righteous, but will remain separate and apart from them, and then its deficiency will be apparent and it will lack the power to rise."

— Benayahu ben Yehoyada, Berakhot 5
Rabbi Chaim Vital:

"He [the Arizal] also told me, when I went with him to Tiberias to the cave of Rabbi Akiva, that Rabbi Akiva told him that I should mention Rabbi Akiva ten times in succession before each of the three daily prayers—night, morning, and afternoon—and by this he would be impregnated into me and help me greatly."

— Sha'ar HaGilgulim, introduction 38
The Rashash:

"He should intend to cleave and connect his living soul with the living soul of the righteous person, etc., his soul with the soul of the righteous person."

Rabbi Yaakov Yisrael Kanievsky:

"And it became clear to me that it helped him greatly when before prayer he made a declaration that he was praying according to the mind of the Arizal, the Taz, the Shakh, and the Magen Avraham, and afterward during the prayer he intended that he was praying in connection with a famous righteous person, and it became known to me that this helped him almost entirely to pray without distracting thoughts."

— Kraina D'Igrata, Part I, sec. 281 in the new edition
Rabbi Natan of Breslov:

"But in truth all the prayers—the righteous in every generation raise them up and establish them, as it is written: 'And Moses set up the Tabernacle,' and they raise every limb to its place, and build up the stature of the Divine Presence little by little."

And in Likutei Tefillot he prays:

"And we direct our prayers and connect all our prayers to all the righteous of our generation. And You, in Your abundant mercy, awaken the hearts of the true righteous of our generation, and give them the power to receive our prayers and raise them before You. And even if our prayer is not as it should be, and is mixed with much refuse, and in our prayer there is not even one word or one letter that is pure and clean, and my speech stammers greatly and my tongue is full of defect, for I do not speak with understanding and the speech is far from the thought—despite all this, Your mercies and kindnesses prevail. And give power to Your true righteous ones so they can raise and lift and elevate all our prayers, refine them and cleanse them of every dross and defect, so that they may rise before You favorably, and from them they will build the stature of the Divine Presence, preparing it, supporting it, and raising it from exile. And raise up the fallen booth of David through our prayers, and restore Your Divine Presence to Zion, and shine Your face upon us."

"And grant me the merit to pray to You in connection with the souls of Israel, so that our prayer should be the prayer of the many. And do not despise our prayer, as it is written: 'Behold, God is mighty and does not despise.' And may there be added and magnified above a great and mighty increase of holiness through our prayer."

Rabbi Avraham son of Rabbi Nachman:

"For all the words of the Sages regarding the above prohibition refer to when one intends thereby to make the one from whom he asks into an intermediary, God forbid. But if his intention is only like the matter of communal prayer, where they arouse one another to ask and pray together over their troubles in general and in particular, for each person is obligated to share in his fellow's distress… then it is very necessary to join together in their prayers in the collective, and specifically with the soul of the righteous and wise one among them. And even so, all are also required to join even with the prayer of the sinners and the distant among them, for this too is part of the necessity."

Rabbi Meshulam Feibish of Zbarizh:

"Nevertheless, the words of his prayer rise by a person connecting himself, by saying: 'I hereby take upon myself the positive commandment of loving your fellow as yourself,' and including himself in complete love with the holy souls of the righteous."

Breslov Hasidim who follow the path of Rabbi Nachman of Breslov connect themselves before prayer to all the righteous, and there is a text they say: "I hereby connect myself in my prayer to all the true righteous of our generation and to all the righteous who dwell in the dust, the holy ones who are in the land."

Rabbi Yaakov Yisrael Kanievsky was asked by his student Rabbi Avraham Horowitz about Breslov Hasidim who mention before every prayer, "I hereby connect myself with Rabbi Nachman," and he answered that certainly Breslov's intention is not that; rather, their heart and intention are only to the Master of the Universe, with Rabbi Nachman's merit giving their prayers strength to be accepted.

Rabbi Gedaliah Kenig composed the booklet Chayei Nefesh; this is a letter answering a question addressed to him regarding connecting to the righteous during prayer, while citing the words of Rabbi Chaim of Volozhin on this matter (brought in his book Nefesh HaChaim), which according to the reader's understanding imply idolatry. Rabbi Kenig seeks to prove that Rabbi Chaim's words do not relate at all to the matter of connecting to the righteous. At the beginning of the booklet there was printed an approbation from the rabbis of the Edah HaChareidis, Rabbi Shmuel Wosner, and other rabbis, both Hasidic and non-Hasidic. Later, a letter of approbation written by Rabbi Zvi Yehuda Kook was published, with additions from the author's manuscript.

What the questioner quoted at the end, from what Rabbi Natan said about Rabbi Nachman—that whoever comes to his grave can be repaired—is very puzzling, and reminds me of quotations from Messianic Jews who believe in Jesus.