the best intuitive argument I think is looking at different cases. The problem with the way scott presented the cases is that you could argue ...

B. The prize is behind door B. The host eliminates door C, you switch to B and win.

C. The prize is behind door C. The host eliminates door B, you switch to C and win.

... that either case B or case C is impossible because we know now that the prize is not behind that door. So there are only 2 cases hence 1/2...

I'd explain it like so (this will also explain why how the door was revealed matters):

Case 1: The door you picked originally was the correct one. In that case, switching loses.

Case 2: The door you picked originally was not the correct one. In that case, switching wins.

This is always true, regardless of how the revealed door was chosen. Now all we need to know is the probability that the door you originally picked was the correct one.

If the moderator purposefully revealed a goat, then it is 1/3, because him revealing a goat tells you nothing about your original pick, because whether you pick the car or a goat, he can always reveal a goat. Again, Bayes.

If the moderator revealed at random, then it is no longer 1/3. It is not immediately obvious why it is 1/2, but it is logical that it has to be different.

Now if we assume he revealed a goat on purpose, then there remains a 1/3 chance that you picked right initially... and therefore a 2/3 chance that you are right if you switch.

But if someone presents you with a reason why it is 1/2 then you can't usually respond immediately because it takes some time to carefully think through every step and figure out where it goes wrong. And if the discussion is offline, then whelp.