Everything else is overthinking.
Yes, it does. this is the entire point behind the popularity of the Monty Hall problem. It is counter-intuitive and feels like overthinking, but it is truth.
I described the situation pretty clearly (imo). I did not need to think about it much at all, though it did take some thought to describe it clearly.
I can tell you, without much thought, having been trained thoroughly in probability and statistics, it is 50-50 if a door is opened randomly to reveal a Goat. You can switch if it makes you feel better - because it is 50-50 so you wont be hurting yourself.
https://en.wikipedia.org/wiki/Conditional_probability
there are other examples that make it more clear why condition matters.
Whether you end up switching or not, the other opened door is 2/3 likely to contain the prize and the door you picked originally has a 1/3. Whether you switch or not, doesn't change the odds.
If you picked the prize P(A = 1/3), then the random open is going to be Goat P(1). if you didnt pick the prize P(B = 2/3) then the random open is going to be Goat P(1/2) or Prize P(1/2)
P(Goat reveal) happens (1/3)(1) + (2/3)(1/2) = 2/3 times
P(Prize reveal) happens (1/3)(0) + (2/3)(1/2) = 1/3 times
P(you picked prize, given goat reveal) = (1/3)(1) / (2/3) = (1/2)