Why shouldn't one just stop making car payments and let the finance company repossess a car one can no longer pay for?

If it comes down to that you can't pay for it then you should call them and tell them to pick it up and repossess. There are consequences:

1. This will be on your credit history, making it difficult for you to purchase anything on credit again.

2. They will sell your car at an auction (often sold for well below what you owe) the remaining balance you still owe.

3. On the balance that you still owe they can take you to court, which can lead to garnished wages.

If you are going to lose your home or can't pay for your food because of a car payment it may be your only option, but you should only do it in those extreme situations. If you financial burden is temporary, talk to your loan company about deferments that will alleviate short term financial stress.