This depends a lot on the school. A research university is going to have a lot of people that students never meet: accountants and lawyers to ensure compliance with federal regulation on grants, for example. The phrase "administrator" also gets thrown around a lot in these discussions without much attention to more nuanced differences, i.e., a VP or vice provost is a very different thing than staff who are not paid all that much. And some people classified as staff do teach -- boundaries aren't always that sharp.

This is not to say university bureaucracies are not bloated, but the bloat is multi-faceted and often grows in different directions for very different reasons.

I think the point is that it's not simply the case that universities have lost sight of their purpose of teaching people, but have broadened the scope of their mission beyond teaching … or broadened the definition of teaching? Broadened their mission? Certainly made their mission less focused/clear. And this leads to bureaucracy. See Harvard's Mission Statement.

https://college.harvard.edu/about/mission-vision-history

in loco parentis writ large.

They need more bureaucrats because they’ve grown dramatically in scope. Schools used to be primarily about education. Now they’ve become Club Ed. A 4-year vacation paid for with unsecured loans. Full of gyms and climbing walls and aquatics complexes, libraries and recreation centers and football stadiums.

Where did all these buildings come from? Wealthy donors and government grants. What do all the bureaucrats do? They run all of these facilities, their maintenance staffs and kitchen staffs and counselling staffs.

This is the reason all these costs have gone up: nobody donates money to pay for bureaucrats. They donate the building and saddle the school with the maintenance bill. These are white elephants that do nothing for education. Schools like them though because they increase their marketability. Taken as a whole, these things are a huge liability for society because schools are in a zero sum game when it comes to competing for students.

But Universities have always had that kind of stuff. At the uni I went to, the sports facilities, fields, pools, gyms, tennis courts etc. are all run by a non-profit society that had been operating for more than a century. All the food outlets were owned and run by the Union (although more are being outsourced with chain franchises appearing now), not the University itself.

The corporate bureaucracy was growing, but it didn’t have much to do with those student services or facilities because that was all separate.