The Monty Hall Problem is a probability puzzle, but the actual problem with the Monty Hall problem was just Monty Hall. Probability doesn't really matter if the game show host has free will and a willingness to mess with you.
This is the classic 'three doors' problem, where there is a prize behind one door and nothing behind the other two. After you, the contestant, picks Door 1, the Host then opens Door 2 and shows you it is empty. The host then offers you a chance to switch your choice to Door 3? Do you take it?
If probability were the only thing at play, yes. However, as Monty Hall noted: if the host is not bound by any rules, they can mess with you and are not predictable.
Monty was the problem, hence the name of the problem.