the people in power aren't stupid so why are they backing this?
I'd argue that they're not stupid, but they are very, very busy, and preoccupied with a huge number of other issues that they're supposed to be informed about, and they regard this one as relatively unimportant, so they haven't been informing themselves about it in any depth. Which of course has to be true of a lot of issues that people with power deal with, just because there is so much to deal with and know about.
They link it to gay rights, and see it as similar, something that will gain them credit for being caring, modern, open-minded conservatives. And why wouldn't they, with Stonewall using their huge popularity/ respect to insist on that link? Most of them will be unaware of the actual facts, and just accepting that if Stonewall is backing it, it must be OK.
And don't forget that there are lots of lesbian/gay/bi people who are conservative voters: sexual orientation has a strong innate component, so there's no reason to believe it's linked to political affiliation. So they'll be thinking about that as well.