I don’t necessarily think gotcha is the right word, but I know what you mean. I think it often belies a lack of confidence in the feminist justification and is often the first or only example of women’s need for single sex facilities.
There’s a belief that people who argue that women aren’t entitled to single sex space might be more willing to consider it if it has a patriarchal basis.
The subtext is: “oh I see what you mean about needing single sex space now - you don’t want her imam, dad, future husband, etc, thinking she’s unclean/damaged goods, because a bloke has seen her hair, maybe in those cases single sex spaces should be allowed”.
It’s a case of using the ‘masters tools’.
I know it is always couched in the language of inclusion, but I get the strong instinct that its over-emphasis is coming from the same place as people beginning every criticism of gender identity ideology with “first I’d like to say, I believe in trans rights” or by amplifying the voices of male transsexuals over women. It’s feminists pre-emptively conceding to misogynist opposition to deflect potential criticism.