Depth perception is a fusion concept. We don't acquire any depth information directly from the senses (other than touch), but it can be inferred by comparing the results from each sense. The brain actually uses many sources of information. Stereographic information, trying to pair features from one eye against features in another eye, is one source of information. Depth of field is another. And, as you and artists have noticed, shapes can be used to convey depth even when painted on a flat surface. We can even use our ears to sense depth. If we notice that a person is speaking, the shape of the reverberations and the near/far field effects of the sound can be used to assign a depth to that person.
As you noticed, closing one eye removes one source of information that is telling you that a scene is flat on a wall. That can encourage the brain to emphasize the other sources of depth, such as apparent vanishing points.
As for why we don't watch movies with one eye, that's more of a psychology question than a physics question. I can venture a guess that the goal of a movie is to be immersive, not merely to have a sense of depth. Closing one eye might help with depth but would make it feel less immersive.
It's also worth noting that there's plenty of other hints suggesting to the brain that the scene is flat. In particular, it's presented on a wall, and we all know what a wall is like. As a counter example, people do set up non-motion flight simulators, where the construct a fake cockpit, but it remains stationary -- there's no hydraulics moving it up and down. The screens on the outside of the plane "windows" are flat, and not 3d. However, in that situation, you have many hints telling you that a scene is "real." People who make such flight simulations often hear that pilots can't believe how real it feels, even though there's no actual motion. And said pilots certainly keep both eyes open while using the simulator.
