I might be boring, but I would hate a climbing gym first date. It’s public and easy to leave, but requires certain clothing, involves weird contortions and angles, and I’d get red and sweaty af. I’m not especially femme (so no makeup problems), but that’s still a recipe for self consciousness for me.

Obviously if you float it beforehand and they accept, that’s great (and they’re probably more what you’re looking for), I would just not suggest that as an extension of a coffee date. I could see it going either way from your comment, so I just wanted to note it.