I think the car analogy is most fitting, and even supports Rolex's position.
I used to race cars in the SCCA and saw similar manufactures claims, as related to warranty / service. So you can go out and and buy a Porsche. You can chip it, replace the exhaust, change suspension components. Porsche will void your warranty on the spot. (In fact VW did this a few years back at an import car show. A rep was walking around taking VIN numbers to VOID their warranty).
So, to the point. Do you still own a Porsche. Yes.
To your specific example. ETA in a Gen Datejust because you can't/won't replace the defective 1570. Is it still a Rolex? I won't argue with you. Rolex will.
Lets get philosophical now. If you have an organ transplant-- liver (like Steve Jobs did recently). Because Steve Jobs has an aftermarket liver-- Is he no longer Steve Jobs? Will God (if you believe) void his warranty.
If you really want to get into the philosophy side of when something is gen or ceases to be itself check out: The Ship of Theseus, also known as Theseus's paradox, is a paradox that raises the question of whether an object which has had all its component parts replaced remains fundamentally the same object. http://en.wikipedia.org/wiki/Ship_of_Theseus
Enjoy,