I didn’t understand how this could be permitted even under pressure. Isn’t there a prohibition of “do not come near” here?

Because it is not clear that this is included in the prohibition. The question is whether it leads to sexual immorality, or at least arouses forbidden thoughts. And this is how Maimonides writes at the beginning of chapter 21 of the Laws of Forbidden Relations:

“Whoever has relations with one of the forbidden relatives by way of bodily contact, or hugs and kisses in a desirous manner and derives pleasure from physical closeness, is lashed by Torah law, as it says, ‘not to act according to these abominable practices,’ etc., and it says, ‘Do not come near to uncover nakedness’—meaning, do not come near things that lead to sexual immorality.”

Notice carefully: it speaks of closeness “in a desirous manner.” True, that is the definition on the Torah level, and on the rabbinic level there may perhaps be room to prohibit even without that. But even then, we are still speaking about closeness that leads to sexual immorality (even if right now there is no desire in it). And that is the straightforward meaning of the words of Rav Padda in the Talmud, Avodah Zarah 17a and Shabbat 13a; and see Tosafot there and there.

And also well known are the words of the Ritva at the end of Kiddushin:
“Everything depends on Heaven’s view. And so too the Jewish law is that everything depends on how a person knows himself. If he sees that he needs to keep distance from his inclination, then he should do so, and even looking at a woman’s colored garments is forbidden for him, as stated in tractate Avodah Zarah 20b. But if he knows himself and knows that his inclination is subdued and subject to him and does not stir impurity at all, then he is permitted to look and speak with a forbidden woman and to greet a married woman. And that is the case of Rabbi Yohanan in Bava Metzia 84a, who sat by the immersion gates and was not concerned about the evil inclination; and Rabbi Ami, before whom the maidservants of the emperor’s house went out; and several rabbis who conversed with those noblewomen; and Rav Adda bar Ahava, of whom it is said in Ketubot there, that he took the bride on his shoulders and danced with her and was not concerned about improper thoughts, for the reason we have stated. However, one should not be lenient in this unless he is a great pious person who knows his inclination well, and not all Torah scholars trust their inclinations, as we see in our passage in all those incidents that are brought. Happy is the one who overcomes his inclination, whose toil and trade are in Torah, for words of Torah sustain a person in his youth and give him a future and hope in his old age, as it says: ‘They still bring forth fruit in old age; they are full of sap and freshness.’”

And all this is regarding an ordinary woman. Here, since we are speaking about someone who will be his wife, there is more room to be lenient (especially if this is perceived as part of the ceremony / occasion). But as I wrote, all this is only a very pressed leniency, and in my opinion it is not proper to act this way in practice.

The opposite, no? On the contrary, with someone who is going to be his wife, there is a clear and established dimension of attraction and sexual desire, all the more so at the marriage proposal itself, where it is basically being declared that this dimension is going to be realized. (In general, it is hard to imagine a person hugging and kissing the one he has chosen for his heart and not feeling closeness and intimacy—which also have a sexual element.)
So even if one is lenient like the Ritva regarding other women in cases of polite touch and the like (even if it is hugging and kissing, as in certain cultures), it is hard to extend that to the present case.

One can raise arguments in all directions (and also argue about them in all directions. Compare his wife when she is a niddah versus an ordinary forbidden woman). You asked a halakhic question and received an answer.