I find that first intuition odd. What if we alter to situation to not involve the new parts? Simply disassemble the old car parts and reassemble them, is that a brand new old-car?
Or how about we make all the new parts plastic so that no functioning "car" is produced by the gradual part-for-part substitution. We end up with two cars, one an all-plastic car model, and one consisting of all the old parts assembled and functioning. Which has the better claim to being the old car?
Yeah, part exchanges quickly leave me with even less salient intuitions about the proper use of old identity terms than these more "splitting" oriented cases. What should we say if a lion talked and all that.
Yes, it has all the parts of the old car, and only those pieces, but it was assembled anew from zero.
A more difficult dichotomy would be: what if we exchange parts between two cars until each one is the other one. That's hard to decide.