One reason that college sports became important in the United States is that athleticism and liberal knowledge became intertwined through the ideology of "muscular christianity." Basically, it was a philosophy that believed manliness, sports, discipline, and self sacrifice were essential aspects for the modern Christian. The YMCA was a huge proponent of this philosophy, and the YMCA was tied to alot of elite New England schools in the 19th and 20th century (ie Princeton, Yale, Amherst.)

Also the period that you point to (early 20th century) is when the United States made a series of imperial conquests that put it on the world stage in a way that it hadn't been before. In 1898, the US occupied the Philippines, Puerto Rico, and Cuba, ushering a new vision for the American male leader as manly, christian, and soldier-like.

that's just one idea, though I'm sure there are plenty of other contributing factors.