This is the same explanation I arrived at too. Crucial to answering the problem correctly is understanding that the host doesn't reveal a random door, but a do

This is the same explanation I arrived at too. Crucial to answering the problem correctly is understanding that the host doesn't reveal a random door, but a door that you have not chosen, and also is wrong. And then the reasoning follows naturally to the correct answer. It's amazing to be so prejudiced that someone would be adamant on the incorrect answer, when it can be explained in 3 simple lemmas.