I think it depends on the situation.
I agree that a kid should earn the RIGHT to use a car. I don't think that necessarily means that they have to pay for the car. For example, let's say my daughter turns 16 and has $1,000 saved up and wants to buy a car. All she can afford is going to be a clunker and I would not be comfortable with her driving that type of car.
My response would be that I would buy the car for her, but she has to earn the right to drive it - by paying for gas and insurance and taking care of it. That way, I can be sure she has a safe car to drive that is less likely to break down on her.
I think that either your method or mine can work - the critical issue is whether the kid sees it as a gift (or, even worse, an entitlement) or whether they understand that they have to earn it.
But, in either case, I don't think any parent has the right to expect the other parent to do something that's not spelled out in the divorce decree. My ex is constantly expecting me to buy things and it's unfair and annoying. If a parent wants to buy a car for the kid, they can ASK the other parent if they'd like to contribute, but shouldn't expect or demand it.