Yeah, the Davidian Revolution, certainly South of the Highland Line, never happened. For example Robert the Bruce - Earl of Carrick, later King of Scotland - belonged to a line founded by another Robert Bruce [there were a lot of Robert Bruces] an Anglo-Norman among many brought North by David 1st to add muscle to his reforms and rewarded with land and the Lordship of Annandale. This did not make them uniquely Scottish though. King Robert's father, another Robert Bruce, was almost certainly an Essex boy, born on the family estate in Writtle. It is possible King Robert was also born in Essex, though more likely Turnberry Castle the seat of his Mothers family; what is certain is it was the location of his second marriage, to Elizabeth de Burgh - the de Burghs being another Anglo-Norman family, established in Ireland. The Comyn family, King Robert's rivals for power, also had Anglo-Norman roots.
