Ya I get that, conceptually it is a logical conclusion for many to make. But once you start looking at the different pieces it is maybe less so. The biggest part is the tax deduction of donations. As an individual you cant offer that, then the potential liability or headache of "I gave Susie $$ but it looks like her parents arent spending it for its intended purpose". Plus its harder to go into a business and ask if they would like to donate to my daughters gymnastic expenses, versus support your local club. The best that I have seen is organizing as a social club or 501c4 which keeps the organizations tax free status but eliminates the tax deduction for donations. But as a social club you can charge membership dues and that can take the form of fundraising X amount or inkind work contribution (working a meet, etc). There is a little more overhead in reporting for this type of organization, but you in theory have more flexibility to allow families to fundraise for their own child versus the group. I know a lot of high school bands booster organizations are organized like this.
But again, the practicality of this is that the amounts we are talking about generally don't reach scrutiny level, so many "clubs" just do what they want despite what the regulations say.