[XBOX ONE S] It seems that any vehicle parts that become ruined on the vehicles in the game, will not give players the option to remove from the car anymore.
I crashed my Olga after a huge lag spike. I grabbed parts to replace the ruined parts on my car, but it wouldn't even give me the button option to remove the parts off of the car.
The radiator, spark plug, tires, and hood were ruined. None of them were removable. The battery was Badly Damaged, but was eligible to remove from the car. So I assume that any parts Badly Damaged or above can be removed from the car.