This sounds crazy...It's not the fact that it's CRT, It's the scanlines, really. Does it have scanlines?
Well my HD tvs are perfect for modern gaming..I have a Trinitron, the kind that was extremely common 10 years ago, and I can barely notice the scanlines and the games still look great. If by CRT HD tv, do you mean something like this:
If you have to game in fantastic quality, consider investing in the Framemeister.