One thing that helps IMO here is to exaggerate the problem:

Start with one million doors.

Choose one.

Now have Monty open 999,998 of them.

Do you switch doors?

I believe it works because the chooser is forced to reckon with the 1 in a million chance that they chose the correct door. Additionally, it becomes clear that Monty actually fed the chooser new information by opening 999,998 doors.

It may not take the first time, but given a few rounds of the million door challenge both truths should sink in for the chooser.

Perhaps it would help to have a Javascript version that starts with a sufficiently large number of doors the first round, then cuts that in half each round until you get down to 3. The odds of winning decrease each time but the principle should still hold. (Although I can imagine some people getting back down to three doors and suddenly reverting back to their previously held intuition...)