There is a real question that lies at the core of this issue. The yeshiva and its administration are here to serve whom?
If the mission is to provide jobs to those seeking careers in chinuch, then the rebbi must be kept in his position at all costs.
If the mission is to provide the best chinuch possible to the children, then this rebbi must be replaced.
What makes this problematic is that we have a situation here of אחד בפה ואחד בלב. No yeshiva would openly say that they are here for any other other purpose than to do the best for the talmidim. But that becomes an obvious decoy for the mission to keep themselves in business, with the “best” talmidim, the “best” image and reputation, and the record of doing such great jobs of hiring faculty.
In reality, there are rotten apples in every bushel, and chinuch is no different than any other employer. If a grocery store hired a worker that was ineffective, that employee would be replaced. There is no acceptable excuse to not replace a rebbi who cannot do the job.
There was a book published in 1969 called The Peter Principle. He studied the subject of incompetence at great length and depth. The result was the formulation, “In a hierarchy, every employee tends to rise to his level of incompetence”. It is a good, healthy read, and highly recommended. One of the tenets is that when the incompetent reaches that level, he/she is maintained there and not let go. I bet this has been witnessed by many. I have observed it in chinuch, but similarly in many other places, businesses, government, etc. Sadly, our yeshivos can boast a good measure of this.
Yes, it is a menahel’s dilemma. But the primary responsibility is not to the rebbi, but to the flock that is losing out on his continued employment. I would insist the stated mission for the yeshiva should dictate the direction of the decision. And upon the notice of termination, the rebbi should be advised to make a change in his career direction.