A few years ago I would have completely agreed with your point about social skills. I don't anymore. Social skills make no difference in the kind of situation described in the OP.
Let me explain...
~10 years ago, I had a lot of social anxiety and I read that the best solution is systematic desensitization a.k.a. exposure therapy + social skills training.
So, with these goals in mind, I forced myself to spend 2 hours every day, just traveling to & fro on public transport and talking to whoever ended up next to me. I did that for ~2 years, before I started going to clubs.
My bid at conversation was always the same: notice something about the other person, other than their looks (Eg: their clothing, accessories, electronics like phone, or their actions), give them a compliment on it, and then ask a question about it.
In the beginning, I would visibly shiver with nervousness and stammer and stumble over my words. But, my plan worked: over time I got more confident and my social skills improved.
So, why do I say I disagree with you?
In those 10+ years, I have attempted to start conversations with 1000s of men & women, but all of my successful attempts have only been with men i.e. every attempt to start a conversation with a woman began and ended the same way: I made my bid, got either ignored or got 2-3 words answer, and then "hinted" that the conversation was over... and I took that hint.
Sure, many guys acted that way too, but with 1000s of bids under my belt in those 10+ years, I have had conversations with 100s of men. But, not a single woman.
Why?
Was it the way I looked? No, I have received several compliments on my looks, a few compliments about how I dressed, and I maintain good hygiene. So, I look good enough.
Was it what I said? I tried multiple ways to start a casual conversation, and even went to clubs so that I could try out more direct non-casual ways to start a conversation with a woman: it went worse than before, and casual "openers" did as poorly in clubs as well.
I did end up parting with random men a few times, though.
After eliminating various other possibilities, I was just left with one: visible nervousness.
While I did get more confident in talking with men, because my bids often did work with them, I was still pretty nervous when approaching women because in spite of 1000s of attempts I haven't had a single positive experience.
When you have social anxiety, "just change your mindset" doesn't really work that well. Without at least some corrective positive experiences, the anxiety will still continue to be on full display.
And, as you said:
I can absolutely tell the difference between a guy who just chats with everyone vs a guy who is exclusively trying to get the courage to ask for my number.
I have a feeling most (all?) women share this mindset: He's nervous while talking to me? He definitely wants to fuck me.
And, as u/polyrhetor said:
if the only thing they think about communicating is “hey I find you attractive” then they can fuck right off. Like, is that my only role? To “be attractive”?... Being told someone else has assessed your attractiveness and found you acceptable before you even sit down is gross.
In other words, for a man, nervousness / social anxiety / not having confidence = creepy.
And, of course, women are disgusted by creepy men.
So, if you attempt to talk to a woman and you aren't born confident, she'll be disgusted that you even tried. Your social skills don't make any difference at all (in fact, you can't even build your social skills when it comes to women) because if you look nervous, no conversation will go beyond a few seconds.
So, is learning social skills useful?
For women, yes.
For men, yes, for talking with other men; but for talking with women, it makes absolutely no difference.
So, if you aren't already confident, pack up, go home, there's nothing you can do. Get a pet for companionship.