But when I watch a movie, the “black” that I’m seeing in a particular scene isn’t the absence of light, because it’s not actually “black.” It’s a very very dark shade of grey or brown or whatever. And that requires light.

What you’re seeing are the deactivated TV pixels absorbing light. This doesn’t work with a projector screen because the screen is of course designed to be reflective, otherwise you wouldn’t see anything. Point a projector towards a piece of black velvet and you see… black velvet.

Even if there is some actual “black” (spots where no light is coming out of the projector), there will still be a gradient, and immediately after “no light,” you will have a light attempting to project a very dark shade.

This is the contrast I was referring to earlier. It’s basically the accuracy of the projector in defining a limit between the areas it’s lighting up. But if you do this in a room with the lights on and the windows open, the image will be completely washed out regardless of how high the projector’s contrast is.