Let's remember that there is a lot more to universities than undergraduate education.
And university sports should just be professionalized. The big-time sports are a cartel designed to avoid paying young athletes while still selling tickets to watch them play, and the small-time sports like the one I played could be moved to public or private sports clubs.
Why are academic education and sports playing so closely tied together?
Most of our football players for example would be a complete waste of a seat in any academic institution.
I think big-time college football and basketball may be pure historical accidents. The sport of American football had its earliest success as an Ivy League college sport; back when the rules were being firmed up Harvard was the national champion. Wikipedia:
In the late 19th and early 20th centuries, gameplay developments by college coaches such as Eddie Cochems, Amos Alonzo Stagg, Knute Rockne, and Glenn "Pop" Warner helped take advantage of the newly introduced forward pass. The popularity of collegiate football grew as it became the dominant version of the sport in the United States for the first half of the 20th century.
And basketball was literally invented by a school sports instructor as an indoor sport suitable for play in school gymnasiums:
http://www.kansasheritage.org/people/naismith.html
... and the inventor immediately joined the University of Kansas.
So basketball and American football may have been born and raised in schools, which explains why they're historically tied to schools in their native land.
My impression is that the other big-time American college sports are late additions to an existing big-time college-sports system. For example, baseball grew up with a complete minor league system operating independently of colleges, and college baseball has only become big within the last few decades. And ice hockey is a regional college sport in the USA but in Canada, where they take it seriously, it's a club sport with a hockey team in every town larger than a critical size. (My impression is that the critical size may be as small as "six people". ;)
I'd go so far as to say that actually teaching students things is the least important service performed by universities even for undergraduates. Far more important for students' future careers is signalling their performance over the course of three years of relatively autonomous study within a controlled environment of their choosing, and the commitment that entailed. Not to mention the huge variety of extracurricular opportunities offered that are particularly hard to replicate through the medium of a website.
Performance could still be measured by grades even in a purely online university.
Extracurricular activities are still possible, people could join study groups within their local area and socialise there, possibly also discounted travel to visit with study groups elsewhere. In addition you could spend the money you saved via cheaper college fees to indulge in some pretty kick ass hobbies.