The problem is that it is not obvious why the generalisation mus be that the host opens n-2 doors. Many people view it as the host opening just one door, and doing so also in the 1e9 case.

What complicates things is that these people are right, too. The generalisation of the problem does not fix the number of doors opened by the host. It could be that the host opens just one door. Or 5e8 doors, splitting the set in the middle. Or any other number in between. In fact, the number of doors opened by the host could be decided by dice before each round starts.

So now the person you talk to goes, "I'm still not convinced, but let's say for the sake of the argument that switching is good in the situation you mention. Now tell me why it is better to switch in all other cases."

You're ending up in formal proof country a lot faster than you'd want for an intuitive explanation.

(Odd observation: if k is the number of doors opened by the host, I find any case where n=2k+1 a lot harder to explain than when k=1 and n is some large number. Maybe that can give some insight into why it's a tough problem.)