Is gaming even possible on a cheaper ($100) secondhand projector?

Hai!

I'm thinking of replacing my cheap crappy 1080p sony TV with a projector. The only usecase will be playing zelda on docked switch. It's not a platformer, but I'm spoiled and want decent latency (20-30ms).

How is the latency of 1080p-but-still-cheap projectors on the second hand market?
Is gaming ever a possibility? Or should I stick with a TV?

What should I look for? Specific brands or product lines known to be good for latency?