My friend uses his PS3/PS4/Xbox One with a projector, a high-end Epson @ 1080p. The only reason he's not too crazy about it is that the Epson is limited to 60Hz, so he was considering a 4k television. Not until the prices come down a little though. He set the projector up on a second floor, which is facing an open area above his living room, using a 10-foot screen which is set onto a fold-out metal frame. Since there was no decent way to split the sound, he bought an expensive receiver and set up speakers all around, which don't really get in the way (much).
Now, there are two reasons I wouldn't recommend a projector, namely the limited lifespan of the lamp (lamp, not "bulb") and the heat generated from the unit. Scratch that, make that three reasons, considering the sound output solution you would have to create. There are fairly affordable units made by ViewSonic, BenQ and Optoma, and one ViewSonic I found on Amazon had audio out, via a headphone jack. It's not the best for sound, i.e. two channel stereo, but at least it's there.
The lamp life on home projectors has also increased significantly, as I remember a time when the lamp life for a $700 projector was around 300-400 hours, so the lifespan has increased dramatically. Granted, the lifespan of a projector lamp is nothing compared to an LED television.
Another option to consider would be a short throw projector. Unfortunately the only short throw projectors I worked with were at my last job, and they were all Epsons (BrightLink), meant to be installed on an arm directly above the screen. I imagine that home projectors would have good keystone adjustment should you have it on the floor instead.
