(First: a disclaimer. Someone will try to make a bad-faith interpretation of this as me opposing gender-neutral restrooms. I'm fine with gender-neutral restrooms, but it's intellectually lazy to point to them as a solution to "the trans bathroom problem.")

In what universe is it easier to convince the population to completely give up gendered multi-user restrooms than it is to simply let trans people use the restrooms of their gender, as we have been for generations?

Do you know how trans people typically choose when to switch which restroom we use? We typically do it when we start getting weird glances or comments in the restroom of our birth sex. In other words, trans people pick the path of least disruption or resistance. You switch restrooms when remaining in the old one would be more disruptive than moving to the new one. Most trans people go out of their way to cause as few waves as possible. All trans people need for this is to just be left the fuck alone. We're handling it. Just mind your own damn business and stop legislating our lives.

Compare that to what you propose. You need to get the entire society onboard with getting rid of gendered bathrooms. You need to get parents OK with the idea of their 10 year old daughter sharing a restroom with a 300 lb 6' tall man. (And not just bigot's stereotypes of trans women, actual big cisgender men.) I'm sorry, but while maybe in an ideal enlightened world, that would be possible. But we just don't live in that world.

Trans people: "Please just leave us alone and quit getting into our business."

Enlightened centrists: "No! We'll keep the bigoted law in place restricting your bathroom use. But no worries, soon we'll just abolish gendered restrooms all together, and the problem will be solved!"

With allies like these, who needs enemies?