At least in the Once and Future King, the only real King Arthur story that I've read, I got the impression that Arthur pulling the sword from the stone was more of a metaphor of him "being ready" to be king more than just genealogy or anything like that.

When Arthur pulled the sword out of the stone, he was remembering all the stuff that Merlyn taught him about the different ways that animals run their societies and how it informed how he would lead if he were in charge.

That might be TH White's flavoring to it though.