I can only assume the king actually wanted to get rid of the waste of skin prince, but for publicity reasons it needed to happen honourably. So, the king encouraged the prince to conquer dungeons. Knowing the prince is useless, the king expected the prince to kick the bucket soon enough. Alas, the MC kept the prince alive. So, the court mage became the new problem. That's why when the prince fired Alek, the king didn't mind at all, even if firing a court mage would obviously be up to the king, not a mere prince. Now with the court mage gone, the king's wish of the prince getting himself killed might finally happen. It will be a more or less honourable death in battle, in the eyes of the population and foreign nations.