A lot of the time, when we "try to improve our social skills", we are consciously trying and change/control the things we say and do, whereas the things that people are responding to are things we have no conscious control over or even awareness of, such as our eye movements, facial/hand/body tics, posture, breathing, and other very subtle signals in the way we speak and engage with others.
Years ago I spent quite some time trying to consciously improve my social skills - including the kinds of techniques taught in the "pickup" community back in the mid 00s when The Game came out. It got me nowhere, and I hated how manipulative it felt.
Later I discovered some subconscious emotional healing techniques. I've been undertaking them consistently for nearly 10 years, and things just keep getting better steadily. I haven't worried about consciously trying to be better socially, other than just generally being polite and attentive to people, but my relationships have all improved.
(That said, I've also become a better programmer, and better at working in teams, as I've become less anxious and better able to work through challenges and obstacles without getting flustered).