Yes, those other organizations have the same exempt status. My guess is the NFL makes more than all of them combined.
The thing you're not getting is "the NFL" doesn't make a lot of money. The 32 pro football teams that are represented by "the NFL" -- all of whom pay taxes individually -- make a lot of money.
When you're talking about "the NFL" that does not pay taxes, you're not talking about the teams; it is the league office, basically. Any money that comes into the league office is used to promote the industry of football -- basically the business of its 32 tax-paying member teams. The $9billion TV contract people hear about: "the NFL" doesn't get that, the 32 teams do and pay taxes on it.
This is exactly the same as a Chamber of Commerce taking in some money and using it to put out a cool brochure about local businesses. All the money coming into the CoC is used to promote a local industry, the members of which will then pay taxes on their increased business. It makes no sense to tax the CoC because it is not a for-profit entity. Similarly, "the NFL" head office is not - in and of itself - a for-profit entity. It exists to hope to maximize profits for its industry (the 32 teams) all of whom pay taxes.
And the amount of money "the NFL" has that it doesn't pay taxes on is by one recent estimate, less than $10million. Which is dwarfed by thousands of other non-profits.