The more I read about this topic, the more I find it to be a red herring in the internet echo chamber with a reach all the way to the floor of Congress. Sheesh.
It would appear some individuals are using the tax exempt red herring to leverage greater disclosure regarding compensation in the league office. Evidently, some think the guys there are overpaid. Well, the owners are paying them out of their own pocket via those annual "dues", so it's their problem, or at worst the union's problem to whatever extent it affects the salary cap.
I've not yet seen anybody demonstrate the tax avoidance savings.
In fact, the MLB league office had the same status as the NFL league office, and ditched it in lieu of a taxable LLC structure. This was actually noted by the lawyer who wrote the source piece for the OP blog piece, citing a source by footnote that the tax consequences are non-existent, but then questioned the veracity of that source with no support! Do you think MLB did that to pay MORE taxes just to avoid certain disclosures? I think not. This seems to boil down to crankiness over executive compensation.
Here's another gem from the lawyer's original law review piece. He notes that one of the league office's key functions is to make 0% loans to the franchises for stadium improvements, as though this is some kind of scam. I don't know about the 0%, but it's beside the point. The league office financials show $1 billion in borrowings, $800 million in loans to teams, and an interest expense of about $50 million. So, even if the loans to the teams are at 0%, the owners are collectively paying about 5% interest on those borrowings through the league office, paying that interest via their dues. If anything, the function of the financing arm of the league office is to spread the financing costs across all teams...stadium financing as a collective endeavor. That makes sense...a large chunk of gate receipts are shared.
Until somebody walks through an example of how this tax exempt arrangement saves owners on taxes, I'd be highly skeptical.