I can't give any insight to the OP's question, though I do have a comment.

Scott is also a family name that is of fairly ambiguous origin. I have been told there are at least three different origins (English, Irish, and of course, Scottish), none of which are related to one another. By some published accounts, the Scottish line traces back to one Uchtred filius scoti - Uchtred, son of (the?/a?) Scot - in the early 1100's, witness to the foundation charter of the abbey of Holyrood , among other things.

The Irish and English Scotts presumably derived their family names from ancestors who migrated from Scotland and either referred to themselves as 'Scots' or were referred to by their neighbors as 'the Scot'. (e.g. 'Alex, the Scot' turning into 'Alex Scott' over time). The second 't' was added later as spelling standardization took place, to differentiate the family from the nationality/ethnicity.

Regrettably, my family doesn't know for certain which of the three groups we descend from. A cousin has traced back to pre-Revolutionary western Virginia, but the trail gets a bit confused trying to go back from there. From what I recall, when asked my grandfather always said we were Irish Scotts, then laughed. I think he was making a joke based on the fact that his mother was Irish (from Country Antrim) and the family name is Scott, though there is a distinct possibility that we are, in fact, of Ulster Scots descent. As to where those Scotts may have came from, who knows?