Ah, the paradox works :D
No its not an independent choice because the initial state depends on your previous choice.
You have a 2/3 chance to choose a goat initially, the host is guaranteed to eliminate the other one, when you switch then you will win the car. You have a 1/3 chance to choose the car initially, if you switch then you will lose
so if you switch you have a total chance of winning of 1×2/3 + 0×1/3 = 2/3