I have a projector, a BenQ X3000i, in my living room, with a retractable screen. It has the plus side of not needing a dedicated wall, but does perform poorly (vs a TV) if the room isn't darkened. Maybe eventually I'll tie it into my home automation with some smart curtains.

It has low latency, will do 1080p 240Hz, 4k (pixel shift) 60Hz and HDR. Can even do 3D content if you really want...

BenQ did include an Android TV stick in the box, but you can just not hook it up to the projector - problem solved.