I think projectors are great. In fact, I currently have one. But there are lots of trade offs. They’re big and take up lots of space, especially the good ones. Placement can be awkward even if you get a short throw, unless you ceiling mount, which isn’t always practical. Relatedly, it can be a pain to hook up to sound because the projector is in the back while you need sound from the front. Image quality can be decent but is still way worse than pretty much all modern TVs. (I hear laser projectors kinda fix this but they’re even more expensive.) It doesn’t turn on instantly; there’s typically a significant warm up period for the lamp. Some units have a noisy fan because the lamp produces a lot of heat. You need a large clear wall space or a rollable screen. I think there’s a reason why projectors are typically in movie rooms and not for more casual spaces.

All this to say, projectors are great but not for all contexts. I wish the decision to get a projector and the decision to get a privacy respecting device were two completely unrelated decisions.

I thought PC monitors would be higher priced than commercial displays, but I haven’t really looked into it. It sounds like I should get a pihole either way.

It didn’t cross my mind that I could run Linux on a tv. (I figured, however, that the pre-installed software is built on Linux.) Are you talking about something like LinuxTV.org