The biggest single thing to keep it going would be it is a symbol of power. Kings learn and perform magic. This likely has become an expectation even for the masses. If their king can't perform any magic then 'we need a new king who does!'. This might also work into a 'strong' king to protect their own countries. Even if all the king can do is light one candle with extreme effort.
Even small acts that no one else can do can add to the mystique of power and the right to rule. Being able to light a candle without flint and steel can be very useful too!
Now why would it be mandated by the empire in the first place? My best guess would be they control the magic school, they can know the strengths and weaknesses of the ruling kings and can control what is learned. Those that are very powerful, (or might be) are either recruited to be strong allies or quietly put down and blamed as an accident that happened during class. Magic is dangerous you know...
Also while 'teaching' the children of the kings the empire is 'protecting' them and indoctrinating them. Since as the next king, you wouldn't want anything to happen to them would you...