There are good answers already for why does there continue to be a huge emphasis on sports in American academia but none really answer the question.
The fact is that sports in America were introduced at universities out of necessity. Where in most parts of the world there has been long traditions of clubs or the local handling of games/sports, America had nothing. One small town might play another small town in a "sport" but that didn't satisfy everyone. You had elitist or exceptional athletes that wanted to compete against their equals, not Gary the blacksmith.
So this is mid 19th century and America is boiling. A nation divided on many subjects. So instead of a local rowing club or in today's terms playing for your company team, the easiest thing to gravitate to is a local university. They had the money, organization, place to play the game, and so on.
And back then universities had opinions and power concerning government and policy. So the elite universities (most were in this group at the time) wanted to take their debating and add physicality to it. Races, rowing, simple games. It invoked pride and if Harvard won the rowing competition then they must be right about slavery.
I didn't even ask who has time for games in mid 19th century? Well you are probably a male, somewhere between 20-35, you have lots of money, and no job - you go to school. This is the epitome of sports culture. Where are all of these people stacked at... Universities. So it was just the perfect storm.
Now once it started the early collegiate sports scene really was much like we see today - except it was admittedly like that in the late 19th century and early 20th century. What do I mean? Well players were old. You might not have many players on your football team under 20 and a few in their 30s. Some players student-status was highly questioned. There weren't really any rules at first and when they started the rules in the late 19th century there were ways around them.
Players were paid, sometimes "pros" went back to college, there were boosters... the schools were driven by pride, power, and money. Maybe the only things different were (lack of) media and that they were not preying on teenagers.
And the evolution of sports in the 20th century has gone from we have money and power so we will form the best teams, to we will get money and power from having the best teams. The big D1 schools are the worst. They hide huge huge earnings by allocating costs to sports teams so they can make millions/billions on tuition and licensing - yes everyone buys Texas Longhorns shirts for their Economics department.
Some universities "claim" to be losing money. There have been economic impact studies done showing that almost none that made the claims were even near losing money on sports. When they factored in advertising, enrollment, exterior sales, and so on. Really the only thing that makes sports somewhat costly for universities now is Title IX. Very few women's sports make money and most women wouldn't go to a university because their softball team is good.
So now we have the NCAA, colleges, tied-in businesses getting profits and tax breaks for players that are playing for free. This may change now that there has been talk of unionizing but could be years and years down the road.
Even if this happened and the landscape changed were the big sports went to a club system there would still be sports in American universities. They would function because students expect this now. Things would probably work like they do for club sports at current universities or how things work at most DIII schools. You play local teams, you drive to the game, pay for your equipment, maybe offset a little by entrance fees or a nice booster.
So why are there sports in American colleges? Pride, money, free-time of students, and the fact that there weren't other organizations to handle these things in the new America. Why will sports be played in American colleges in 100 years? Same reason they are played at clubs in France. Tradition.