While these empires didn't have any say in deciding the succeeding principal of the Master Teacher Academy, they did have a vote of confidence. If it was found that there was an overwhelming lack of trust in the succeeding principal, they could launch a formal petition to the Master Teacher Pavilion headquarters to look into this matter.
After all, the operation of the Master Teacher Academy was largely financed by the surrounding Tier-1 Empires.
The surrounding empires would often donate large sums of money to the Master Teacher Academy each year for their operations, and in exchange, the Master Teacher Academy would nurture their master teachers, thus bringing their national power to greater heights. An incapable principal could render their investment futile, and this was something that they weren't willing to see.
If there was widespread distrust in the capability of the principal, even the headquarters would have no choice but to reconsider the matter.