I have only skimmed some of the material you provided.

I'll reiterate that I don't claim any expertise in the relevant areas of law, but these statements give me pause:

The Naval Academy Athletic Association is not a government agency or instrumentality and is not part of the United States Naval Academy or the United States government. The United States Naval Academy, nor the United States government, has any obligation or liability for the actions, decisions, commitments, or liabilities of the Naval Academy Athletic Association.

My cursory reading of the material and in particular these excerpts on serves to reinforce my non-expert thoughts that this arrangement is problematic, from multiple perspectives.

From a legal perspective, the NAAA almost certainly operating as an agent of USNA, including when it schedules intercollegiate athletic events in which USNA participates.

That strikes me as basic agency law: the agent (NAAA) acts on behalf of the principal and the principal is legally liable for the acts (and omissions) of the agent that are undertaken (or omitted) on behalf of the principal.

The principal cannot avoid its legal obligations by contracting with a third party to undertake those obligations.

But they have been doing it for a long time. Maybe no one had a reason to challenge it but...