Again, it wasn't *pushed for". Trans people have always used the restroom for the sex they identity as, so there was no need to "push for" anything. It was conservative who started making a stink about it only in recent years. Your attempt to flip the script was adorable, though.
They only thing they "push for" is rights, liberty, and equality.
But let's say they did "push for" it. That still wouldn't explain how you would know if a woman was trans or not as they enter a restroom.
Btw, the guy in this pic was born as a female.
Do you think it would go over well if he were forced to use women's restrooms?