I don't think it's that hard.
I mean otherwise Charlemagne blood would not work for Michael - as he is definitely not legal heir to Carolingian monarchy.
I think any royal lineage - not even legitimate in sense of crown would work well enough - so even Y chromosomes would not necessarily work - it could all be through maternal lines. (Which also is a reason why I dislike this idea - because I guess at least half of mankind have SOME royal ancestor which makes whole RULE quite cheap.).
Michael may be descendant of Charlemagne in some way but he is definitely not one in line that would make him eligible to throne of France for instance. Neither are other guys. It's more than mystical element of king's blood stick around for a long time.
Now Aragorn had very fine claim to the throne as last descendant in direct male line of Elendil. But then if we compare it - Knight of the Cross is not a King. In fact probably he should not be a King (though we know there were exceptions) as it would make him involved in various mundane actions that are usually not so virtuous. - But let's compare it to Tolkien.
Sure Boromir, Faramir, Denethor - those guys have king's bloods, High Numenorean blood, thought they are not king lineage. They are not as strong, tall, longlived and willpowered as Aragorn, still they are way more powerful in those aspects for good or bad than any random Rohirrim or Daleman or Haradrim.
At least I hope so - because dammit if Jim's want to suggest Knights need not also king's blood but also legitimate claim, I'm gonna fly to USA, drive all the way from Chicago to whatever Kansas desert he's hiding and hit him with a head three times with some big, complex, academic book about history of medieval kingdoms.