From wiki:
BMI can be calculated quickly and without expensive equipment. However, BMI categories do not take into account many factors such as frame size and muscularity.[9] The categories also fail to account for varying proportions of fat, bone, cartilage, water weight, and more.
Despite this, BMI categories are regularly regarded as a satisfactory tool for measuring whether sedentary individuals are "underweight," "overweight" or "obese" with various qualifications, such as: Individuals who are not sedentary being exempt - athletes, children, the elderly, the infirm, and individuals who are naturally endomorphic or ectomorphic (i.e., people who don't have a medium frame).
One basic problem, especially in athletes, is that muscle is denser than fat. Some professional athletes are "overweight" or "obese" according to their BMI - unless the number at which they are considered "overweight" or "obese" is adjusted upward in some modified version of the calculation. In children and the elderly, differences in bone density and, thus, in the proportion of bone to total weight can mean the number at which these people are considered underweight should be adjusted downward.
Obviously-BMI is more suited for measuring the general population -quickly and easily.....Whereas your suggestion of a fitness test would be difficult to implement-who would carry out such tests? it would just put more of a strain on the NHS,and would not really address the issue of weight in the same way.