Projectors vs. TVs are always a tradeoff in certain aspects. Projectors are amazing and I'm not sure of your size limitations, but 100" is a jumping off point. 120" or larger is pretty easy to hit with today's models and should be about the size worth consideration if possible.
But, projectors aren't exactly like TVs. Projectors need the best room for best performance, and people have an unrealistic expectation that their large white rooms with a big white ceiling is not going to impact their black levels... then they often complain about weak performance.
It's not how it works. So, you just get the best it can deliver for the room you are putting it in. The cheaper projectors, like the BenQ HD3550, which is a pretty popular entry level 4k(ish) projector is going to have pretty good black levels for an entry level model. But, the room will have significant impact on those black levels.
Better, if you can swing it, is the Epson 5040, which is now discontinued, but may be found on closeout in some locations for under $2,000. Which my be a bit over budget.
Cheaper Epson projectors are great TV replacements because they can have inexpensive replacement lamps. They aren't 4K models at all, but will be 1080p limited. They can have pretty good black levels, but not great, but they will have excellent brightness for rooms with less than adequate light control.
So much of this is about the room. The room you are looking to put this into is, by far, the most important part of the discussion when talking about quality and what you will get and what you should expect.