Sports that have little to do with exercise:
- Racing (Horse, car, hound)
- Chess
- MMA (While they are super fit, it isn't "healthy")
- Boxing
- Bowling
- Sumo (nothing healthy there)
... The list could go on, but I think I've made my point.
The purpose of sport for the companies that make money from it is, well, making money, and that's from advertising, gambling and entertainment, as you say.
But society grants sport a lot of tolerance and even encouragement that it does not grant to other such profit centers. What is the motive for society to do this? Answer: it dates from the time when sport was healthy exercise. Now that the ratchet of ruthless negative-sum competition has turned sport into something that does more harm than good, this social tolerance is highly inappropriate, and it should be revoked.
Panem et circenses.
I have never heard of any rules against professionalism in soccer, or even norms about it. Boxing has always been a sport where the very best were pros. The gentleman's code bit was in the rules against biting, eye gouging or using your legs. Even rugby is at best half true. Rugby split into union and league over professionalism. I believe union in France akways had sh shamateurs as well. I don't know enough about American Football to comment but what I do know about American sports culture does not incline me to believe there was ever a ban on pro play.
Insofar as I'm familiar with a sport you're wrong on all counts except rugby union.
Most of the sports you listed require tremendous exercise though. Especially car racing. Fighting sports... don't know why you list them at all. Even chess requires a broad exercising regimen.