Similarly, imagine a regression predicting student grades, including terms not only for individual students, but also for individual teachers, type of class, time of day, etc. Wouldn't a student's regression coefficient be a better measure of their school ability than a simple GPA?
I can't be the first person to have thought of or suggested this. Regression aren't rocket science, and millions of people are now running their lives around GPAs and student evaluations. So I can't help but conclude that the people who could make this change really don't care much about how good GPAs or student evaluations are as measures of student or teacher quality.
(Soon after writing this, I learned that in fact many people have made similar suggestions. See here and here.)