It's also annoying in a semantic way.

"At all costs" sounds like it's saying to take every sacrifice possible, even if not every cost is needed.

"At any cost" sounds better, because it's saying something is important enough for whatever cost is necessary, but don't make sacrifices that aren't needed.

Well that's the cost they're hoping for. The costs they aren't willing to pay are things like changing culture to be more harsh towards men who don't respect consent, listening to victims of sexual assault, or punishing powerful men who hurt women.

And compare that to keeping trans women from playing a non contact sport with cis women and tell me which will actually protect cis women.