In 100 years, I could imagine a society where young people are expected to find a gender identity in the same way as they find other aspects of themselves, and maybe their gender identity changes a few times over their lives. Imagining such a society doesn't bother me at all. I think it's totally workable on a cultural level, I don't see anything inherently wrong with the idea, and there have been a few cultures in history which have kind-of worked like this. But right now, our culture is very far from working like this, so people who try to live like that today are unfortunately going to face a lot of difficulties.
IMO it's more likely that the current focus on flexible gender identity as an important thing which is and should be relevant to everyone's daily life will be looked back upon in 100 years as a sort of fad which eventually disappeared. That is, 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, for example. I'm not especially rooting for or against this outcome, but it just seems to me that the recent movement on gender identity is mostly pushed by a relatively small group of ideologues, and it's not relevant or useful enough to the average person for our culture to actually permanently shift in such a major way.
Especially in people younger than ~20, I think that transgenderism today is often (but not always) a result of depression. It's a way to escape one's life, imagine oneself as a completely different person, and attract attention. But because it makes life 100x more difficult, it feeds the depression rather than helping it. This, I believe, is why the suicide rate among transgender people is so high: it's not primarily because they're harassed for being transgender, but because they were clinically depressed from the start. (This is just a guess which I have no evidence for.)
There are also people who are not depressed, are thinking perfectly well, but just prefer the idea of being the other gender. Some people are born with genes which give them poor athletic performance, but that doesn't mean that they have to like it or give up on ever running fast or whatever. With effort alongside modern abundance & technology, it's possible and IMO downright virtuous to fight against nature to get what you want out of life. It would be widely considered offensive today (for understandable but ultimately irrational reasons), but I do think it'd be similarly fine for someone to prefer to be another race and work toward being recognized as a member of that race.
The most important thing is for everyone to display tolerance and compassion. If you act hatefully toward LGBT people, or exclude them when you could include them without much cost, then you're being an asshole. But when LGBT people get angry at people for accidentally "misgendering" them, or when they intentionally try to put people into difficult situations as some political statement, etc., then they're being assholes. The most difficult situations are where there is no solution that will make everyone perfectly comfortable, like the question of which locker room transgender students should use, and in these cases it's especially important to treat everyone involved with compassion, not dismissing anyone's feelings as bigoted or unreasonable.