And suddenly, privacy invasion gets off the table quickly (as soon as you are not aiming for 100% non-cheaters, the cost becomes obviously too high for everybody else).
As far as reputation, I'd rather see schools focus on the successful students, which is somewhat done with all those research-paper-grading systems (not a perfect system by any means because of gamification, but at least idea in the right direction), but mostly done with bragging about scientific break-throughts to come out of their students and staff.
Still, what is the purpose of a reputation or "well-respected degree"? The goal should be knowledge and applicability of that knowledge to actual problems in life (known as "jobs"): it's not like anyone accepts any graduate without interviewing them first, which is to say that nobody trusts any school to have done a proper job of evaluating them. Most of those schools don't trust themselves, so they hold interviews for post-graduate studies too! :D
Nobody looks at the "lemons" coming out of a school to consider it a bad school (I am sure you can find plenty from "top" universities too), but on the successful ones. Do the successful ones change with more cheaters at all? (Sure, there is a turning point, but catching all of them is meaningless)