I think MM had 100% control over coaches underneath him. I think most, if not all, coaches do. Personally, I cant see why anyone, worth a ****, would become a HC without being given that authority. Your livelihood, reputation, and legacy hang in the balance. It's up to you to establish a staff you can work with and have the same philosophy.
Sure TT could approach MM and suggest a change, but it maybe a decision that TT would have to fire MM as well to get rid of Capers. I think MM is worth keeping. I dont know why MM held onto Capers as long as he did. That has been his biggest error as a HC.
Some posters, not talking about you, dont have a clue what no talent or poor coaching looks like. I am reasonably sure that TT was demoted due to health reasons and not because of performance. Otherwise, he would have been fired completely.