But providence too is something we are not supposed to grasp, at least in terms of the outcome, if souls are reincarnated, no?

After all, with all due respect to the rabbis who announce to all of us that they understand “why the coronavirus came to the world” (for example), a more serious statement, and a more characteristic one, is: “We do not know the calculations of Heaven.”

Could you clarify the difference between “something we are not supposed to grasp” (the World to Come) and something you call nonsense?
What is the meaning of something we are not supposed to grasp?

I don’t understand the complication here. These are very simple things. The World to Come or reincarnation of souls (that is not the same thing) are theses that nobody has any way to test. Information reaches us from the Holy One, blessed be He, about such a thesis, and the only question is whether that information is reliable (that is, whether the Holy One, blessed be He, really said it) or not. In contrast, with providence we are talking about a claim regarding a change in the laws of nature by the Holy One, blessed be He. That is something we are supposed to see, and the fact that we do not see it is a refutation of that thesis (maybe there are excuses, that He hides from us and the like, but the burden of proof is on the one making the claim).
None of these is nonsense. Nonsense is meaningless talk (not talk that is incorrect).

There is an explicit Mishnah: ten things were seen at twilight on Sabbath eve.
And the things are listed there that we see as miracles.
That is, according to the Mishnah, these too were created already at the creation of the world, meaning they are not miracles.

Whereas regarding the World to Come there is no dispute among our rabbis about its very existence [there are explicit Talmudic passages], the issue of reincarnations—whether this is true or fantasies taken from superstitions that began to become widespread in the world a little over 1,000 years ago—is the subject of a fierce dispute among the medieval authorities (Rishonim) [the Geonim rejected it, and if I am not mistaken even scorned it and mocked it], and even later there is dispute about it. And who can decide? Especially since it is not mentioned at all in the words of the Sages, neither in the Babylonian Talmud nor in the Jerusalem Talmud, even though the Talmuds did not refrain from including all kinds of wondrous, strange, and fanciful aggadot… which implies they did not know the issue at all…

“We are talking about a claim regarding a change in the laws of nature by the Holy One, blessed be He. That is something we are supposed to see, and the fact that we do not see it is a refutation of that thesis.”

The desire to see God performing the miracle
under the microscope through which you are observing is doomed to failure.
God performs His miraculous acts when there is a need for that, not when you are forcefully looking for Him.
And trying to find Him retroactively will not work either, because the miracle hides within the regular natural chain. You cannot track it at the particle level.

It follows that the whole thesis that says “we are supposed to see Him intervening in nature” is incorrect from the outset, and it is not exactly clear what style of “divine intervention” it is arguing against.

In order to place the burden of proof on us, we first need to hear a claim that truly describes God’s mode of action.

This is the usual, widespread apologetics, and I have explained more than once why it has no basis.
Divine involvement in the world means a deviation from the laws of nature. We are not talking about an expectation of divine involvement at a specific time and place. If you examine statistically, for a group and a control group, the dependence of medical and other conditions on prayers or on observance of commandments, that is an empirical test in every respect. And there is no great problem in doing that.
About any law of nature you can claim that perhaps we did not see it, and that not seeing is not proof. That is not how science works.
In short, whoever wants to dig in behind the accepted example can always do so. But it is not reasonable, and in my view even the person doing it does not really believe it. I discussed this at length in several places here on the site and in the second book of the trilogy.

What about the question that even if the information passed down from the mouth of the prophet until now is reliable,
there is still your question as a Kadmon-like argument, which asks that since regarding the World to Come there is not much reason to convict the wicked, because they no longer exist anyway, then there will be no Gehenna or anything of that sort.
I don’t remember exactly. But something like that.

“If you examine statistically, for a group and a control group, the dependence of medical and other conditions on prayers or on observance of commandments, that is an empirical test in every respect. And there is no great problem in doing that.”

I am committed to the truth.
I assume that if we take 10,000 terminally ill religious believers (with months left to live) over whom many prayers were said and much charity was given and so on, and we take 10,000 terminally ill atheists over whom nobody prayed, we will discover that neither of the two groups survived.
Maybe there will be one or two religious people who recovered, but probably we will also find one or two atheists who survived.

If so, then apparently your question is correct—what is the meaning of prayers? Does one not necessarily see here that there is no divine intervention?

As I understand it, after the verdict has been issued from Above, something really, really major is needed in order to change it.
The judgment itself, in my view, is almost always not natural but rather a judgment from Heaven, whether it is because of something the person did in this reincarnation, in a previous reincarnation, or simply because he has finished his part in this world.

Think of a righteous and pious Jew, one who gives charity and does good, who somehow ended up committing murder.
The judges sentenced him to life imprisonment.
Immediately they brought countless advocates on his behalf, and people begged the judges to change the sentence and take into account the righteousness and commandments he had performed. So many people told how he financed their son’s surgery abroad, and how he would study Torah for hours in the study hall, and so on and so on.

Would the judge change the sentence?
Probably not. And rightly so.
There is such a thing as sweetening judgments and so on, but the judgment fundamentally still exists, because it is known that the defendant committed first-degree murder.

Why are there so many rabbis and sources showing that one should pray and God will save (“even if a sharp sword is resting,” etc.)?

A. Because of the not-so-high chance that something very, very significant really will happen and God will indeed save.
B. For the feeling of the person praying.
C. The prayers will go toward something else, not toward what we are directing them to. Maybe something connected to that person.

This is not apologetics on my part, but an honest attempt, even if painful, to present Jewish faith and clarify that the Holy One, blessed be He, does not work for us. In order to change His will, to change a decree that has been issued, something truly major has to happen.

Let us emphasize that if “reincarnation of souls” did not exist as a fairly basic concept in Judaism nowadays, then indeed your question would be stronger, since one clearly sees that there are people who suffer without having sinned.