Although I definitely see where you are coming from, private schools do not select students solely based on athletic ability. Generally, if one has the necessary money and grades, they will be admitted. Many kids go to private schools for the education and are far from athletic, yet these kids still count towards a school's multiplier.
I also would have to say that the financial aspect is a challenge, rather than a benefit, for private schools. Private school tuition is extremely expensive, and so are the personal trainers that you mention. Many parents do not possess the financial means to afford this, and this significantly cuts down the pool of kids from which a private school can draw.
Also, I cannot see how it is a disadvantage for public schools to have to take students who are on welfare. One's financial standing does not correlate with athletic ability.