This is true for Bayesian updating on any information, and is a good lesson for life in general.
If he can remove an unchosen prize door with some unknown probability p and does not reveal the contents of the removed door, then the probability of getting the prize by switching is simply (2/3)(1-p).
His motivations and knowledge just amounts to some distribution of p, but that’s a rather different problem.
When the expected value of p is 1/2, then switching is equally likely to win you the prize as your original choice, but the probability of winning is still only 1/3 in either case. Only when p=1/4 will the remaining door contain the prize with probability 1/2.
The missing piece of the puzzle that seems to frequently be left out is we KNOW Monty's motivations, knowledge, and how he decides. At the time the problem was formulated, when Let's Make a Deal was still on television, all this was common knowledge: Monty is the host of a game show, he knows where the two goats and the car are, and he decides which door to open by picking one of the unchosen doors he KNOWS has a goat behind it. The problem suffers now, 40-odd years later, by the fact that people often don't know who Monty Hall IS, and the problem is usually worded as if it's still 1963-1976, so this foundational knowledge is missing. There is no QUESTION what Monty Hall knew, or what his motivation on the show was... except people who don't know the game show don't know that.
Granted, the actual game show was a bit more varied, but the presumption in the thought experiment is that Monty ISN'T trying to goad you into un-choosing the correct door, which he frequently did on the show because the tragedy of it was more entertaining than just giving away a car.