If I look at the way my relative is treated, women assume they are a threat, and men assume they are a cissy/traitor, so they are treated worse than a non-trans person who is biologically male. As I said, because they see trans as traitors, men won't let my relative use the men's toilet.
As a biological woman I at least don't have to worry about whether there will be a toilet for me, because if there is one for men then there will definitely be one for me.
Also, this idea that we shouldn't treat transwomen the same because they experience different stigmas to cis women. As a non-white women experience different stigmas to white women, as do disabled women, but yet ethnic minorities like myself and disabled women aren't denied the right to identify as women because their experience is slightly different.
Men can have less privilege than women in multiple specific situations.
A white, able-bodied, straight man will have more privilege than a white, able-bodied straight woman.
But a white, able-bodied women will have more privileged than, for example, a black, disabled man who is mentally ill, or a white man with a severe developmental disability.
So women do usually have less privilege, but there are specific situations when that isn't the case.