Completely right.
Not only are they not a business, but even the vaunted "athletics programs" have not been operated as for-profit businesses - that is, not until the networks crammed billions of dollars into their collective educational g-strings.
Many college programs operate at a loss. And even in the vaunted SEC, the majority of the money is reinvested directly back into the program, in the form of facilities, athletic support, coaching salaries, travel, debt servicing where new facilitie are being built, etc. The amount of actual profit generated by college programs has been, at best, negligible. When a program is very good, yes, the school that runs it get to bank some small degree of money, but that's it. Which is why I chuckle whenever people go on about how "they've made themselves rich at the expense of the players." Aside from the bloated salaries for football and basketball staff, and the ADs at the major programs who pay them, which I'll granted have been Jimmy Sexton'd into the stratosphere, ain't nobody getting rich. All that money was sunk back into the programs, to create opportunities for current and future athletes - because the mission wasn't profit, it was to sustain and grow athletic opportunities for students at the school. A trhiving ecosystem of student-athletes competing at major P5 schools.
Well. At least, it was. Not anymore.