You are solving the wrong problem (correctly, but still the wrong problem). Coins have no memory, but this game does. The car goes behind a door once. It is wherever it is no matter how many doors are opened. It is not a sequence of independent coin flips.

This scenario has inspired arguments for decades. It will inspire arguments for more decades. It is unintuitive. But I assure you, my original explanation is the correct one.

hmm

when there were 3 doors, you had a 1 in 3 chance of getting the right one.

but eliminating the 3rd door does not carry over to the next question.

there are now 2 doors, the car can still be behind either door.

eliminating the 3rd door earlier is not part of the problem anymore.

it is a new "bet" for door 1 or door 2

*because he is allowed to change his guess.*
now it comes down to how the question is worded (imo).

if you ask "would his odds of having chose the right door improved if he chooses to switch to door 1?"

the answer to that is YES, because he chose BOTH door 1 and door 2 - but he didn't win yet.

but the OP says:

You should now select door #1 to increase your odds of winning.

BUT with the NEW bet, he only has ONE choice between TWO doors, so his chance of WINNING is 50/50 no matter which he chooses.

he won the first bet at 1/3 odds.

but he would always win that bet because no matter what is behind the door he chose, there is always another crap prize to be revealed.