>Your math doesn't work out here. I can't major in football
Literally, no you can't major in football. From a practical standpoint many of the athletes who go to a school like Alabama to play football do end up majoring in it. Maybe their actual degree will say something sports specific like kinesiology or sports management, maybe their academic advisers will point them to general education classes specifically targeted for athletes and they will end up with a super generic business or communications degree, but either way football is the primary focus for many of them.
>From my experience, club sports and clubs such as the theater club are funded via student fees, whereas departmental projects such as drama and music are funded via tuition.
The point is who funds a play or a concert at universities? Do they all need to be financially self-sufficient or do we accept that these pursuits have value outside of the number of people who will pay to watch them?
>In theory, the major areas are self-funding because of the tuition costs.
This isn't true. The costs to teach different subjects varies wildly and departmental budgets are therefore not always inline with the number of students in each specific program. For example, tuition from an English department would likely go to help subsidize a more expensive Physics department.