Divine Patronage
A religious aspect could give your world a very different feel. It might not be what you are going for, but I feelthink that it is still worth mentioning.
Magic could be, or could at least be thought to be, the mark of a deity's favor. The empire therefore requires that any rulers they have can demonstrate that they are powerfully blessed in this waytheir reign is sanctioned by the empire's deity. After all, what better way to ensure the deity's blessing of the land than to put one of their favored ones on the throne?
This would most likely prevent those who are particularly weak in magic from being considered as heirs to the throne regardless of their lineage. Given that magical capacity is genetic, monarchsrulers would probably want to ensure that their spouse is also magically capablegifted to give their children the best chance atgreatest likelihood of being able to inherit the crown after them. To maximize their chances at proving that they have the deity's blessing, all possible heirs would need to be rigorously trained in magic. Disagreements regarding the ascendancy might even be resolved by a magical duel, because surely the one the deity prefers will be the victor!
This isn't to say that lineage needs to be meaningless next to the importance of powerful magic, though. It would most likely be a mix of both, at least partly because only the relatively wealthy could afford to have their children trained in magic in the first place.
Dominant religions can be quite long-lived (as shown by a number of real-life religions that have lasted thousands of years), so this social structure could possibly last the required millennium once it's taken a strong hold. If the deity involved is genuine and at least somewhat active, then it would be even more likely.