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You also didn't explain why the answer that the contestant should switch doors is the correct one

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Of course he did. He explained that staying give you 1/3. So changign give you 1-(1/3) = 2/3. 2/3 > 1/3. End of analysis of this very complicated problem...

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No, he didn't. Here is what he said:

"The answer is: Yes

The odds of the person initially making the right pick is 1/3. So if he makes the switch, he'll be right 2/3 of the time."

This is simply a statement of the correct answer, with no accompanying explanation as to why it is the correct answer.