The permission for cooked wine in the Talmud comes in addition to the earlier decree of libation wine, and cooked wine is exactly the same thing as wine on supermarket shelves, and like gentile cooking in a Jew’s home, which the Raavad permits, etc…
Regarding whether the reason is strong or not: the very existence of some small boundary, whether through cooking or through ordinary Gentile wine in a case where the gentile himself brings it to you, prepares it, and pours it, is enough as a reminder of social distance between a Jew and a gentile.

Does the Rabbi agree with the points I wrote?

Again, thank you very much

I do not know whether “His Honor” agrees. But I am inclined to agree, as I wrote to you. Except that there is a problem with permitting something even when the reason has fallen away, and therefore I would rule that way mainly in a case of need. There are many decrees that no longer really apply today. Lest one repair a musical instrument, grinding medicinal herbs, and so on. And there the reason does not apply at all anymore, and not only in a particular situation (as in our case with a bottle sitting on a supermarket shelf).

Seemingly, the Sages agreed on this, since they wrote that an infant does not render wine forbidden as libation wine and does not forbid the wine for drinking, and the Rema also cited a community that wrote this and defended them somewhat. What you wrote, that there is a problem permitting even when the reason has fallen away, does not apply here, since the Sages fenced the prohibition and cannot bring all the possible examples.
Am I correct in these points?
Seemingly, in the case of cooked wine they should have forbidden it exactly like wine owned by a gentile, and not only in a case where he prepared it for you or where you are drinking what he personally brings you, no?

No, as I wrote. In the Shulchan Arukh it says that one should not read by candlelight even if the lamp is two stories high and there is no concern that he might tilt it. That is despite the fact that according to the basic law it is permitted to read with a watchman present. Here too, you could argue that this permission should be extended to other circumstances.
As stated, I wrote that there is room to be lenient when there is need. I do not see the Shulchan Arukh as binding authority, and you are allowed to think differently from it. But I would not bring proof from the Rema for this (and he too only offers a defense, not a permission), and I am not confident enough to permit it across the board.

No, as I wrote. In the Shulchan Arukh it says that one should not read by candlelight even if the lamp is two stories high and there is no concern that he might tilt it. That is despite the fact that according to the law of the Talmud it is permitted to read with a watchman present. Here too, you could argue that this permission should be extended to other circumstances.
As stated, I wrote that there is room to be lenient when there is need. I do not see the Shulchan Arukh as binding authority, and you are allowed to think differently from it. But I would not bring proof from the Rema, and I am not confident enough to permit it across the board.