Generally there are no schools for Magicians. It is customary to either be taught at home, or by a master. There are some schools, made for Wizards and Witches, but not many students choose to go there. It is because schools don't separate the elements and thus they don't teach properly, don't give enough knowledge.
This is why Wizard and Witch children tend to choose to travel alone and find a Master for themselves. Masters are well known across the world. They have the tendency of rejecting students who don't show enough potential or whom they think won't be able to keep up with their teachings. This is when the student can choose to go home and learn from their parents, or go to a magic school.
Magical talents generally show up before the age of 11 and thus magical students tend to be young.
The reason why Oderia accepts students from the age of 14 is that so they have some time to adjust to their abilities. The school is now well-known in the Magical world and students with special talents are sent there. If one student does not have these special powers, that just means they have not found a way to use them yet and hence, they do not know about it.