"I'm not sure, Allan, that the Royals can actually trace their direct British lineage back very far, at least only by making some sideways leaps"

There is no great sideway leap to be made. The idea that they aren't British comes from the Hanovarian succession. The present monarch's ancestor Sophia of Hanover was declared heir to Queen Anne's throne. She was born in the Haig right enough but she was also the daughter of Pricess Elizabeth of Scotland who was born in Scotland and also happened to be the daughter of James VI of Scotland and I of England. This wsn't a great leap. Charles I was younger than Elizabeth but he was made king because he was male. Those were/are the rules but Elizabeth wasn't any less of British heritage than he was. Sophia died before taking the throne so it then went to her son George I. The present monarch can trace a single British line back more than a thousand years with only a small couple of generation blips where they lived outwith Britain. Not to mention all the other British ancestors she has.