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The original game in Let's Make a Deal went exactly like you described. Monty Hall opened the remaining doors; no option to switch.

Then there is the ambiguity as whether the host will always offer you to switch or just when he hasn't revealed the car.

Deal or No Deal does this - the contestant selects a briefcase, then opens up briefcases randomly. It's very similar to the Monty Hall problem, with some twists iirc.

Yeah. That is literally the whole reason to use the game show analogy.

Why not? Reveal all the choices the contestant didn't make, one at a time, slowly increasing the odds of a prize. This adds to the suspense. It's used all the time.