I'm sure all this is already decided, but I was thinking that one way you can fictionally avoid all these breeding mechanics, is to have the channeler not be the actual sovereign, but actually a respected, immortal servant of a normal royal house, kindof like a Merlin type character. The kings think that the struggle is all about them, but really it's a battle between wizards.
If the channeler needs a king for legitimacy, then all of a sudden you have a reason for sons, daughters etc.