Definitely TV, not likely to go the projector route. A TV is not included in that budget and is being considered a separate purchase. That said, it looks like the budget would, preferably, be $2.5k-$3k and would likely include the following. A receiver, speakers and a video playback source. As noted above, if they're as capable as most good DVD players, a game console would be a great solution.
Regarding speakers, having read a few threads here, I see a couple have left out subwoofers. I don't see myself doing that but I'm very open-minded about it and need to learn/hear more before making that call. Front Center/Right/Left and rear Right/Left are essentials.
Regarding TV, I'm a fan of Sony. They just appeal to me. Definitely something in the 4K family. I haven't spent enough time in the house to determine the screen size just yet.