They also are some of the best too. My wife went to a top-tier school and I remember the controversy when it was discovered that almost 50% of the tuition was actually given to the medical school. The justification was that they were a top school, they would charge what the other top schools charged, and they had a moral imperative to make the medical school as cheap as they could make it. They also guaranteed (in writing) that if any graduate had difficulty paying off student loans, the alumni network would step in and make the payments.
At my Uni, the administrators taught bullshit courses on personal development et cetera.
Law schools have a unique culture. Teaching requirements are a product of their "first among equals" attitude, not a cause. Porting the mechanism without the culture is unlikely to solve the problem.
This is where the education industry can learn from the law schools. They are certainly ivory towers, but part of that tradition is that law school deans still ussually teach. My first year contracts prof was also the school dean. Law school deans remain largely 'first among many' rather than a separate profession, the 'professional administrators'.