I don't think that Andy Burnham likes women. He is a man's man, and I think when he thinks of how he appeals to "voters" he is not even thinking about women.
He completely talked down to Yvette Cooper during a previous leadership campaign debate, which put me off him for life. Regardless of what you think about Nicola Sturgeon, he was completely disrespectful to her position during covid, also.
I have to admit though - I can't get worked up about public toilets. I would far rather share female toilets with transwomen than unisex toilets with men, which is where the alternative is heading. The two incidents I can think of locally with attacks in public toilets were 1) a woman being raped by a man in female toilets - the sign on the door didn't stop him. 2) a young boy being raped in male toilets. It just doesn't seem safe for women, transwomen, transmen or children to have to share toilets with men. In an ideal world, there would be ample unisex "other" facilities, for families, trans people, accessibility issues not covered by a blue badge, etc in additional to male and female only, but it's just not pragmatic in the short term at least.