But by not switching doors, you are ignoring the new information that the
prize has to be behind one of the other two doors....

No, it doesn't. That's not correct.

You are sticking with
your original guess that had only a 1/3 chance of being right. By
switching
doors, you are including the new information that the prize has to be
behind
one of the other two doors, and your new chance of winning is 50%.

No, it doesn't. Your new chance of winning is 2/3.