Why does the Monty Hall problem not apply when the contestant picks the second door?

What matters is that in the case where the contestant didn't pick the prize door initially, the host uses his knowledge to avoid picking the prize door to open.

On the other hand, if the contestant picks a door to open at random, there's a risk that it's the prize door he picks. When that happens, he doesn't get any useful chance to switch -- and this only happens in the case where the original rules would have led to switching being a benefit.

Thus, letting the contestant pick the second door would remove some switching opportunities that would be beneficial, but would not remove away any opportunities that would lose.

Therefore, the chance of winning by switching becomes smaller overall: It drops from $2/3$ to $1/2$.