You're right and wrong.
The MLB doesn't "own" baseball nor does the NFL "own" football. Both of them are gestalt entities comprised of the member clubs.
The NFL are the 32 member clubs. MLB are the 30 member clubs. You can't start a football team and compete in the NFL because the 32 member teams don't want to play against you.
Any concessions a team makes "to the league" is really a concession made to the other teams. For the NFL, every year, the 32 owners get together and vote on various rule changes. Same with the MLB.
MLB is a little weird in that it does have a government allowed monopoly on professional baseball, but no other league does. Like, you could start a rival baseball league, but MLB could take whatever action it wanted to squash your league (assuming all those actions were legal otherwise). But nothing except very anti-competitive practices are stopping you from starting your own baseball league. Just, good luck airing your games, or finding fields that can seat more than 500 people, or being able to sell tickets online, or advertising. MLB can make agreements to exclude rival leagues from everything.
The NFL can't do that. Which is why you get the USFL, the XFL, the Spring League, the AAFL, the XFL again, Arena football, etc. It's just that no one is capable of putting up the money to compete with the NFL. You're either overpaying what any NFL club would pay for a player or fielding players no NFL club would take. And that's not to dismiss any of those guys in terms of athletic ability. Being in the top 1% of athletic ability is still pretty fucking good. But the NFL would be more like the top 0.1%.
But no one is doing that. Average salaries for all of these leagues were under the average salary for the NFL of the time. There just isn't the money because there's no base. And it's because the NFL has built its brand(s) over decades. The NFL makes money hand over fist because they've gotten there over the years. And they essentially got in when the competition was on their level. New competitors on the scene have a much harder path.
The biggest difference is that professional sports leagues are essentially team owned and team run. Collectively, but still.
A better analogy would be Wilson, Rawlings, Nike, Spalding, etc. Wilson make a football called "The Duke". It is made to the specifications set forth by the NFL. They also make the NBAs basketballs. Rawlings makes the baseballs for the MLB. Wilson/Rawlings gets exactly zero input into how the game is played. The people who agreed to play each other do that.
Whereas in eSports, the maker of the equipment (essentially) is the one dictating how to play the game. It would be like if the US Playing Card Company decided to start dictating how the World Series of Poker was run.
So what you have is that eSports is seen as advertising for the game rather than the product itself. That's what separates other leagues from eSports. Every other professional sports league treats the competition as the product. Mainly because they have to. Riot, Epic, Blizzard, Valve, WotC, etc. all see their "professional" leagues as advertising avenues for their "actual" product.