There are tons of women who enjoy the damsel in distress trope and think it’s harmless fun.
There are also tons of women who think it perpetuates sexist stereotypes against women.
Both groups of women’s opinions are equally valid.
Does this not prove that the statement is independent?
Really no, any logic in the real world that has anything to say about damsels in distress is going to be inconsistent, just like the real world is. Therefore it proves everything and has no indepdendent statements. Also, it will have second-order quantifiers so it won’t have that kind of proof theory. If you treat damsels in distress problems as something like knight-knave puzzles from logic, then sure, you can treat them mathematically. But that’s not so interesting.
There is an excellent book you might like, " Gödel’s Theorem: An Incomplete Guide to Its Use and Abuse" by Torkel Franzén, that discusses various forms fo what you’re trying to do, and explains why it doesn’t make much sense in the end. It’s available from the usual places including pdf’s on the internet. Book review: https://www.ams.org/journals/notices/200703/rev-raatikainen.pdf
If they’re all inconsistent then the real world is inconsistent and therefore the real world doesn’t exist since inconsistent systems don’t have models. But all the women who think that the damsel in distress trope is harmless fun/perpetuates sexist stereotypes do exist, and they are models so I don’t think the theory is inconsistent.
No the real world operates by physics which is consistent, but above that are artificial constructs like damsels in distress, that are inconsistent. There’s no world in which those things all work as advertised. We just get by anyway.