Aren't there several small projectors available these days? If you want to do this for whatever reason, it's not like it hasn't been possible for a while now.
I don't really get the point of having a "semi-portable" console in your house. I have zero urge to pick up the big, heavy PS4 and carry it around my house just so I can project an image in the ... kitchen? Some other bedroom? What's the point? I have a nice TV with a nice comfy couch like 99% of people in my living room. No desire to go elsewhere in my home to play. Wii U kinda has the same issue, it's a solution to a problem that no one was asking for.
It's not 1955 anymore, if someone wants to watch a TV show which someone else wants to play a video game or watch the basketball game, most households have 2, actually more like 3+ TV sets.
