Public schools in the US are egalitarian and as public institutions ultimately subject to the will of voters via elected representatives. However, they tend not to be democratic in a sense which would support your thesis in regard to sport.
The standard model for eligibility for scholastically sponsored sports at the high school level is wholly dependent upon were the child lives. Thus High School athletes typically are not free to choose a team which best meets their individual needs.
If the coach basketball coach has a bias against atheists [and in plenty of public schools team prayer is regular and expected] or there is depth at their position or the program simply operates on a seniors-get-first-priority basis, the player is not free to move on.*
In other words, public school sports are largely organized around exclusive geographic franchise rights and there are extensive rules to prevent playing for whom one chooses and severe penalties for doing so by circumventing those rules.
Furthermore, many high school athletic associations restrict how, where, and with whom a player may play in the off season - e.g. by limiting the number of players from the high school team who may play on a club basketball team together during the off season [and that's without considering restrictions on their ability to monetize their athletic fame or talent].
* NCAA rules also severely restrict player movement at the college level - while coaches may switch teams every season.