If gender is an abstract feeling of who you are...
I do not agree with this. When I say that someone is a man or a woman, I am referring to objective parameters.
Maybe that's the root of the problem, then. For a lot of people there is a distinction, gender is the feeling of who you are, and sex is the binary biological either/or.
in 100 years there will still be transgender people, and I hope they are better able to thrive than transgender people today, but it's unlikely IMO that anyone is going to be introducing themselves alongside their pronouns
Yes, definitely. A big part of the problem is the inadequacy of the English language. If we had an appropriate and widely-accepted non-gender-specific pronoun that was used by everyone, instead of sex labels, there would be much less of an issue.