Not every object that has an owner carries with it an obligation of return. The verse obligates returning only under certain conditions, and with regard to a non-Jew’s lost item and something lost in the sea’s sweep, those conditions are not met. As for the issue of unconscious despair, one could answer that something swept away by the sea becomes publicly known, based on what Rashi explained on 22a:
These belong to him, etc. — this is the correct text: “These belong to him because the owners have despaired,” for whenever a river sweeps something away, like beams, wood, and stones, the owners know about it immediately, since it becomes known.

Alternatively, one could answer that indeed there is no permission to acquire the lost item before the owners discover that the sea took it, but one may pick up the lost item and keep it in his house for some time, after which it can be said that presumably the owners have despaired, and then one may acquire the lost item. Since when I picked up the lost item no obligation of return took effect upon me, there is therefore no problem of “it came into his hands through prohibition.”

I didn’t understand. If the ownership remains intact, then there is an obligation of return. Where do you see a disconnect between those two?
If something swept away by the sea becomes publicly known, then perhaps you are right, but then one really must conclude that if there is a situation where it does not become known, it is forbidden to take the lost item. I think the straightforward understanding is not like that. I seem to recall that people discuss whether the despair in the case of the sea’s sweep is actual despair, or whether this is circumstantial rather than psychological despair (the circumstances remove the object from the owner, even if he does not know and has not despaired, and this is so even according to the opinion that unconscious despair is not considered despair).
Rabbi Shmuel’s lectures on Bava Metzia 21a discuss this at length. Note that even in Rashi he writes that in the case of the sea’s sweep the finder acquires it even if the owner does not know.
Here is section 1 from his words there (there is more that follows):

Section 8 — On the law of the sea’s sweep and Maimonides’ view on it
1. In the case of the sea’s sweep and the flooding of a river, even though it has an identifying mark, the Merciful One permitted it, etc. Nachmanides wrote (22b) that the permission to acquire in the case of the sea’s sweep is not by the law of despair, but rather by the law of a permitted lost item: the Merciful One permitted a lost item that is lost from him and from everyone, even without any despair by the owner at all. Therefore it is effective according to everyone even without the owner’s awareness, and so too is stated by Rashba in our passage. [And see above, section 1, paragraph 7, where we brought the view of Netivot HaMishpat, who distinguished between ordinary despair, where it does not leave his domain until it comes into the domain of the acquirer, and the sea’s sweep, where it is a law of ownerless status immediately; and even according to him, the sea’s sweep is a different law and is unrelated to despair.]

However, from Maimonides it appears that the reason for the permission in the sea’s sweep is the law of despair, for he wrote (Laws of Robbery and Lost Property 11:10): “One who finds a lost item in the sea’s sweep, etc., even though it has an identifying mark, it belongs to the finder, as it says, etc.—this one is excluded, because it is lost from him and from everyone; for he has certainly despaired of it.” He also wrote (6:2): “Therefore, one who rescues something from the river or from the sea’s sweep, etc.—if it is definitely known that the owners despaired, these belong to him; and if it is not known, he must return it.” We thus see that Maimonides’ view is that the permission in the sea’s sweep is based on despair. And so too the view of the Jerusalem Talmud cited in Tosafot (27a, s.v. ‘mah’) proves, for it derives the very law of despair from the law of the sea’s sweep. We see explicitly that the permission of the sea’s sweep is based on despair.

This is very difficult for this approach, for our passage explains that the sea’s sweep is permitted according to everyone, even if the owners did not know at all that it was lost from them. But according to Maimonides, that the reason for the permission is despair, there is a difficulty: how can this work according to Abaye in a case of unconscious despair? This is also what the book Nachalat David asks (22a), where he discusses it at length, and ultimately resolves it based on the words of Ritva (22a) regarding Ameimar, Mar Zutra, and Rav Ashi, who came to the orchard of Mari bar Isak. He explained that Ameimar and Rav Ashi ate because it was absolutely clear to them that Mari bar Isak would certainly be pleased that Torah scholars should benefit from his property, and it did not come into their hands through prohibition, for the Merciful One permitted it from the outset even though he did not know, as we say in the case of the sea’s sweep. From Ritva’s words we learn a great novelty regarding unconscious despair: in a case where there is a clear assessment that the owner will certainly despair, it is effective even without his awareness, and this will be explained shortly. And from the fact that Ritva concludes, “as we say in the case of the sea’s sweep,” it appears that he agrees with Maimonides’ reasoning, that even the sea’s sweep is based on despair, and since it is a clear assessment, it is effective even in unconscious despair. If so, one can also say this in Maimonides’ view—that although the permission to acquire in the sea’s sweep is on the basis of despair, nevertheless in such a case it is effective even without awareness. These are the words of Nachalat David. According to this, the wording Maimonides used, “for he has certainly despaired of it,” which in its simple sense implies that he actually despaired knowingly, must be understood as not exact; he only came to explain the reason, that since if he knew he would certainly despair, therefore ordinarily it is considered despair and is effective even without awareness.

I found proof for a disconnect between the obligation to return and the issue of ownership (which affects the permission to acquire):
Babylonian Talmud, tractate Bava Metzia 24b
Actually, it is the Rabbis. Does it teach “they are his”? It teaches only that he is not obligated to announce them. He should leave it, and if a Jew comes and gives an identifying mark, he may take it.
Rashi, s.v. “Does it teach, etc. It teaches only that he is not obligated to announce them” — since the majority are non-Jews, perhaps it belongs to a non-Jew; but it does not become his either, for the Rabbis do not follow the majority in monetary matters.

However, from here we also see that the absence of an obligation to return does not necessarily mean that the owner despaired. So maybe one has to say, like Maimonides, that there is a presumption that the owners despaired of the lost item in the case of the sea’s sweep (because it is lost from everyone), and also in the case of a non-Jew’s lost item: presumably the non-Jew despaired of it, because non-Jews do not generally return lost items, and Jews also do not return lost items to them. But without these presumptions of despair, there would be no permission to acquire it, and all that is learned from the verse is the exemption from the obligation to return.

That disconnect exists only because of the laws of doubt, so it is not a real disconnect. Therefore there is also no need to look for explanations for the sea’s sweep. The doubt is that on the one hand the majority are non-Jews, so it is probably a non-Jew’s, but on the other hand in monetary matters we do not follow the majority, so he cannot take it for himself. There is a doubt here, and therefore the Talmud says he should leave it. But there is no real disconnect here between ownership and return. In the case of the sea’s sweep there is no situation of doubt, so this is irrelevant here. There there is no reason at all to disconnect the permission to acquire from the obligation to return.
In any case, you must remember that you are raising a difficulty (how did they derive permission to acquire it—maybe all it means is that there is no obligation to return it). To raise such a difficulty, it is not enough to show that there is a possibility of disconnecting between the obligation to return and the permission to acquire (which you still have not shown); rather, one must prove such a disconnect. On the contrary, from the Talmud itself regarding the sea’s sweep it is proven that there is no disconnect (from the fact that they derived permission to acquire it, and not only that there is no obligation to return it).