Only if (in fact) it's an independent choice. It's not.
If the prizes had been juggled behind the curtain after one door had been opened, then you'd be right. But the "weird setup", which is the point of the puzzle, makes it clear.
Let's be real here: the point here for most of us is to think "oh it's 50:50", be proved wrong, enjoy being humbled, learn something new about how to interrogate a problem, and enjoy setting this puzzle to younger thinkers in the future.
Not to attack the (very closely-worded) description of the problem because you're upset to be wrong.
It's a probability puzzle and a character test for thinkers.