IIRC it was actually already suggested that trans-women could use the disabled's bathroom, but this was rejected because it didn't validate their identity as women.
Everyone who wants a bathroom can currently have a bathroom. As Caveman says it's about validation.
Trans-identifying men want to be treated as women in every conceivable way (right down to being called for cervical smears and mammograms), which means access to all women's single-sex spaces without challenge. This is
not about any danger or even embarrassment they face in men's single-sex spaces, or they'd be content with the offer of a third space. They reject any such offer indignantly, because it doesn't validate them as being "real women".
We kind of got along for decades because there was a gatekeeping process and women kind of imagined that the creeps and the perverts who wanted access to women's spaces for creepy and perverted reasons were being screened out by that. So even if we felt a bit uncomfortable about it we didn't make a fuss when some bloke wearing a bad wig and a mother-of-the-bride outfit came into the ladies.
Now a few things have changed. First, we discover that we were labouring under a misapprehension. The bloke in the mother-of-the-bride outfit probably still has all his tackle and no intention of getting rid of it. He also quite possibly doesn't have a gender recognition certificate and so hasn't been through any gatekeeping process. This is a concern.
Second, and this is the crux of the matter, the reason the spotlight has been turned on this now is that the militant trans activists have almost succeeded in getting the politicians to legislate to remove the gatekeeping process. Just as women are waking up to what's going on, and saying wait a minute we're not cool with this, they're being told that whatever safeguards there were are to be removed. The excuse is, well people are self-identifying anyway, so what?
Third, obviously the official removal of the gatekeeping and it becoming public knowledge that any man simply has to say "I identify as a woman" and he's untouchable in a female-only space is going to become known to creeps and predators of all descriptions. We're already hearing some reports of creepy men in the ladies saying "I identify as a woman" when challenged.
Fourth, the actual dempgraphic of trans-identifying men is changing. The "old school" transsexuals who just wanted to fit in as unobtrusively as they can are still there, but they have been joined by a band of aggressively male trans activists who seem to take delight in making women uncomfortable and branding anyone who even looks sideways at their five o'clock shadow as a terf, a transphobe and a bigot. This is not OK.
The problem with all suggestions about third spaces is that this solution gives women their single-sex space back. To the trans lobby this is anathema. Women must not be permitted to have any single-sex spaces that exclude males, because "trans women are women" and must not on any account be excluded because validation.
And that's where we are. And if you think that's not autogynaephilia, I beg to differ.