I can’t help but to feel that it’s a bit embarrassing to be playing games like this in my mid-30s in a long-term relationship

I don't find it embarrassing. In fact, I usually get a laugh of those games, especially when the clothing defying physics and holds in mountains with a few strings of thread (Exhibit A). My girlfriend gets a kick out of it as well, but I suppose different folks have different opinions.

To answer your question, most old games are going to bit non-horny. That didn't really come around until the 3D era. It also sells well, so it's hard to find games that don't take advantage of it.