That is a very complicated story with a lot of what I would call path dependence.
At a very high level, one of the distinguishing characteristics of the US higher education system is that we operate a very robust private college and university system in parallel to our public college and university system. In fact the oldest privates had quite the head start, and then the publics really took off after the 1860s with the land grant movement, to the point some privates got absorbed eventually into public systems. But many privates were able to stay independent, and in fact the wealthy class kept creating more privates.
OK, and then in a critical period when college athletics was first taking off in the late 1800s, it was actually the privates that led the way. This all dated back to certain classical ideals of education in which physical education was considered integral, which filtered through aristocratic notions of not just mental but physical superiority, which were reflected at one time in war and then eventually in sport. But anyway, the gentleman scholar was often some sort of athlete as well.
OK, so then the “upstart” public colleges and universities took this up as well as part of their effort to compete with the private colleges and universities for social status and “top” students. And eventually, they grew large enough to actually dominate college athletics. It is hard to imagine today, but at one point Harvard football was 4-0 against Michigan. But that last win was in 1914, and by the 1940s it was a rout for Michigan, and then they just stopped playing.
But by that point (circa WWII and immediately after), college athletics at public universities had taken on a life of its own. There is even more involved with that story, including the GI Bill, the integration of public colleges, and so on. But the idea that a top university would also have a top athletic programs had been cemented in US culture.
But could we undo this now? Maybe, but it is hard to change institutions like this once they are so well-established. Like, Brown actually tried to back down some athletic programs not long ago, and ran into a lot of pushback from alums and such. It isn’t all alums who care, maybe not even more than a fraction, but that fraction tend to REALLY care. Again, public system officials tend to suggest a lot of elected state officials would push back on downgrading their programs, and maybe it is something similar–not everyone cares, maybe not a majority even, but the fraction who do care REALLY care.
Anyway, political issues aside, that is the basic big picture history of how all this happened in the US, at least as I understand it.