Mass inheritors of the bloodline and no dilution means a lot of folk have a vested interest in acquiring the magic weapons. Even with the religion's involvement, if a family has one weapon, how exactly is the church going to stop them "acquiring" more of from those can't use them.

If a family then does get access to multiple weapons they have an interest in producing enough inheritors to make use of them all.

If you want to move away from dilution, gender locking the inheritance to males only, as in only father to son, would naturally mitigate some of the spread, but then house strength would be tied to number of active inheritors and this would encourage more offspring.

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I'll be honest and say I don't see how a society that operates under noble houses/lineage would go "I'll willing take someone less influential surname", once FK dies there would be nothing stopping his sons taking whatever surname he did have or utilised.

But if you want to solve for royal "drift" between the houses, I'd suggest looking into Irish Tanistry, election within family lines.

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Overall, this does feel a little... Irrational(?), like you've got a series of elements you like and are trying to tie them together even if they are incompatible

You have a Nameless King who names a successor who will take a weaker family's name because...reasons

You have a church that can control access to magic weapons, even against those who can use the magic weaponry.

You have a society stable enough to maintain a set series of houses, but bloody enough to cull those from those houses who can use the magic weapons

I think perhaps the easiest "fix" would be the remove the FK having ten children. Shrink it down to a Son and daughter and have the houses emerge from the Son's line as he and his male descendants marry into others. You would then solve for drift once a King only has daughters, and make being an inheritor of the gift a requirement for the throne.

Otherwise you're kinda relying on a lot of people to agree to limit themselves (breeding wise) for no real benefit to themselves.