Other than that, body fat is an indicator not of personal health, but of personal risk.

To rephrase the sibling: if you are obese or overweight, you have, by definition, a certain BMI. (According to the WHO and many national entities, anyway.) The intended common-sense definition of obesity is "accumulation of fat to an unhealthy level", but the diagnosis of obesity is usually based on a statistic which merely correlates with that.

Similarly, to declare that someone is in bad or good health based on their overall shape is to do the same thing: to diagnose someone as having medical problems by observing that they demonstrate a property which often correlates with medical problems.

The sleight of hand is that you're defending a position based on a "diagnosis by correlating statistic" definition of obesity, but you're using the intended definition of obesity ("enough fat that you're unhealthy") to do the actual argument, and relying on the fact that both are referred to by the same name ("obesity") to hide the gap.