I think there's a difference between being healthy/fit and being overweight. There are plenty of people who are technically overweight who can run a mile in 6 minutes. Now, "should officers be allowed to become morbidly obese?" is a totally different question. I agree that at a certain point, you don't need to be a doctor to see that someone's weight could affect their ability to run/chase. If your partner wheezes just getting in and out of a car, then I think it's fair to assume he might not be in optimal shape.
I think officers should have to have a physical, say, every 5 years, and if their doctor thinks they're too out of shape then they should have to re-pass the fitness test they had to take in academy.