Why does the BMI account for height at the square rather than the cube?

Here’s a partial list, which includes disorders normally associated with being overweight, such as heart disease and diabetes. However, that appears to mainly apply if you have a gene associated with being underweight. Even so, having a BMI under 18.5 appears to be as bad as being obese (the lowest risk of death occurs near the high end of the normal BMI range, higher for men than women, according to this study).

What I’m curious about is does a muscular/fit person with a BMI of 30 have the same potential for health problems associated with weight as a fat/sloth with a BMI of 30? In other words, do health problems that are correlated to an obese BMI care what kind of weight it is, whether it is fat or muscle?

The sloth has great health risks; the athletic high BMI indidual has low risks.

Not only does it matter whether the weight is fat or muscle, it matters where the fat is as well. Exercise preferentially reduces the higher risk visceral (aka central) fat. Thus even with the same percent body fat someone who exercises has less health risk.