The professionalization of sports is a short history. And college sports becoming the huge media sensation it is today also a recent phenomenon.
Also, Baseball being the outlier to Football/Basketball when it comes to collegiate leagues probably comes from its very early professionalization relative to the latter two.
Could a single team in other sports go against all the rest in modern times? I highly doubt it.
PS - Rickey was passionate about collegiate sports and the role they played on campuses.
This is partially true. Most Americans still go the NCAA route the typical Canadian route being the CHL, for 16-20 year olds. Europeans follow a model similar to European soccer leagues.
The CHL offers a year of college scholarship for each year played in the league. I think it's only for the national schools, I'm not exactly sure.
Collegiate athletics emerged as lucrative entertainment, over decades. It's not as if they established sports teams originally in order to make money or to become training ground for pro teams. And good luck getting them to forfeit this willingly.
A minor league could show up to compete, but would it offer athletes an opportunity for a rite of passage too? And could it offer a hedge for the athletes who don't graduate to higher pro leagues? IMO baseball is probably a good place to look for answers because it has both a minor pro league and NCAA "student-athletes". Ice hockey, too, IIRC.