Gnu you are incorrect when you say it is 2 doors, 1 choice, therefore 50/50, flip a coin. That is NOT the Monte Hall Game.

The game IS 3 doors, 2 choices.

The kicker is you will NEVER be shown the car OR your first round door (and sometimes it is the same door) on the first round exposure.

Repeat: The kicker is you will NEVER be shown the car OR your first round door (and sometimes it is the same door) on the first round exposure.

So if you follow a switching strategy for the second round (many proofs have been offered so I won't repeat them) the chance of getting the car is 67%. Play the game 100 times expect to own about 67 cars, if the expected of probability holds.

But if you use a second round coin-toss strategy (as you consider the game to really be), your the probability is only 50/50, and YOU CAN EXPECT 17 FEWER CARS out of 100 games, if the expected probability holds.

As I and several others have previously pointed out, no strategy guarantees your winning a car. Chance is a funny thing...you can flip a coin and get heads 20 times in a row, and you could pick the car on the first round 20 times in a row but those are not EXPECTED outcomes.
But a 67% chance is still greater than a 50%, and so you should expect to need a larger garage if you switch.

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Please forgive my use of caps; I am not shouting--I simply don't know how to us e HTML for emphases.