That's how I prove it with playing cards. Except I use a handful of cards, and not a million of them. The trick is in the wording. With 3 you are literally say

That's how I prove it with playing cards. Except I use a handful of cards, and not a million of them. The trick is in the wording. With 3 you are literally saying "let me remove all the cards except for a random one, the one you chose, and the one I chose, of which, one is the correct card to choose. Do you want to change your choice?" So it plays out as expected with 3, and the obviousness of the solution when done with 52 cards.