They tried this a while back and the leagues/Parish's pushed back successfully. I haven't been to a grade school athletic event this year but heard that there are things that they are doing that the AOC asked forAll of these Leagues are private and the AOC doesn't have any legal right or management of the teams/leagues. Having said that the AOC does own just about all the facilities the League/teams use. In theory the AOC could leverage ownership of the facilities but up to this point the AOC has demonstrated they do not want to get involved with the GCYL, GCCYS, WBC, GWAC, CYO, or any of the other Leagues in the area. As long as the folks running the leagues and each individual program don't do anything completely stupid it is very unlikely the AOC will ever get involved in the Leagues.
Here is a link that is on the AOC website
At the end of they day - too much $'s in the Athletic Coffers that the AOC can't ignore.