Why use the flat tire rule when you can just say you didn't realize the time and need to get on the next flight? I was in the lounge not paying attention and literally missed my flight. United just put me on the next flight.
Assuming there are seats available, you'll either get confirmed on the next flight or just be waitlisted.
As for those abusing the "flat tire rule" I don't think it's used that much, I can't imagine...