I'm not American, so your analogies with reference to US racist culture are entirely lost on me.
I know quite a few trans people. Lets stick with the transwomen for now though. I don't think I ever said I had "such great sympathy for them". This discussion needs a little nuance, which you seem to find rather difficult.
Let's take the transwomen I'm on friendly terms with. They don't say or do anything to upset me. In private houses the question of who uses which toilet doesn't even arise. I get on fine with them. But you know, it's possible to be on friendly terms with someone and still think they're a complete weirdo in at least part of their life. That Ashley, or Lesley, or whoever, chooses to dress up in womanface, is part of his personality that I tolerate. It doesn't really make any difference. (I admit that I "supported" Ashley in the past, soon after his transition, like a good little handmaiden, telling him a pack of lies about the softness of his skin and the shape of his torso, because I thought the right thing to do was to bolster his self-esteem. I might make a different choice about that now.) It's part of who they are and I accept that.
On the other hand there are the transwomen I know who I am most certainly not on friendly terms with. They are obnoxious, entitled, bullying creeps. I avoid them, and in particular I make damn sure they're nowhere near if I'm going to the toilet in a venue where they are also attending.
So these are, roughly speaking, the sort of transwomen I know. I don't know any of these ultra-feminine, frightened, vulnerable flowers that some people talk about. I know obvious men who try (more or less unsuccessfully) to present a feminine appearance. Some of them are perfectly OK people otherwise, and some of them are not, but all of them are perfectly capable of standing up for themselves. As a group they are far closer to the people we saw making a spectacle of themselved in Parliament Square than to the more-or-less mythical Hayley Cropper type.
I entirely support the provision of some sort of third space for these people to pee in, if they really can't bring themselves to use the toilets appropriate to their actual sex. Or tacit permission to use the disabled loo, in a small venue. I also entirely support them as regards freedom from discrimination in employment, housing and all other sex-neutral aspects of life. I also support them to be entitled to use the male facilities free from harrassment. And that applies to both my friends and the transwomen I can't stand.
I do not support them to transgress women's boundaries, and never will.