Are tests built into contrib modules a reliable way of testing the success of a module upgrade?
No
Automated tests was created to test code, and to ensure quality of patches. They are meant to be executed in specific, clean environment (e.g. Testbot environment), which your live site certainly is not.
Second reason why it's no is what is tested. These tests only test code. But downloading and unpacking code is simple. Not many thing that can go wrong, and if it goes wrong, you will rather see. The hard and important part is database upgrade - and tests will not try to verify if your data was updated properly.