The problem statement does put constraints on the host, by specifying that the host opened an unselected door with a goat behind it, only to ask if the player wants to change his choice. The answer to the question of whether the player should change the choice is well defined. Other variations are irrelevant since they are different problems.
Your argument is equivalent to denying that 2 + 2 = 4 is correct because the author had the option to write something other than a 2 as an operand.
Nope. The probability theory doesn't work like that. When you argue that 2+2=4 you assume 2 and 2 are known and they are not.
A=you picked the car at first
B=the host opened the door
P(A|B) can be anywhere between 0 and 1.
In your calculations you assume that P(A|B)=P(A) which is correct ONLY if A and B are independent. Independence of A and B is not in the problem statement, you invented this clause yourself.
Your argument is equivalent to denying that 2 + 2 = 4 is correct because the author had the option to write something other than a 2 as an operand.