I assure you, those white kids in the first desegregated schools were sincerely and deeply uncomfortable with the presence of their black peers. The correct response was, and remains, "too bad".
The comfort of the majority is an anathema to the concept of individual human rights.
This comes up so often.
Once upon a time, group A was uncomfortable sharing all sorts of plumbing fixtures (bathrooms, locker rooms, water fountains) with group B. However, society decided that group A's discomfort was unwarranted, and so group A and group B were allowed, indeed compelled, to use the same facilities, whether group A, or group B, liked it or not.
Today, group C does not want to share facilities with group D. Since society once decided that A's discomfort was unwarranted, it must be the case that C's discomfort is also unwarranted.
Of course that is fallacious. We have to decide independently if C's discomfort is unwarranted.
If C is "women" and D is "men", almost everyone would say that C's discomfort is warranted. If C is "men" and D is "women", almost everyone would say C's discomfort is warranted, while noting that, in reality, C isn't all that discomforted.
The whole argument is what if C is "women" and D is "transwomen", because some people say that D is a subset of C, whereas other people say that this is already covered by the case of C="women" and D="men".
However fallacious, though, there is a connection. People note that those who fought for the rights of group A are now hailed as heroes, so people today want to be just like those people, and so they fight for group B, expecting to be hailed as heroes as well. And indeed, some people do hail transgender rights activists as heroes, which encourages them greatly, but not everyone does. In order to make the analogy fit, those of us who aren't real keen on complete and total integration of private spaces for women and transwomen have to be fit intp the pattern somehow, so we are the bigots, the KKK, the segregationists.
It's fallacious, but since when has that stopped a movement?