There are four types of Scots. There are those who descend from Angles and Saxons who moved North into fertile valleys. There are the old Picts who are akin to the original Britons in the South and in Wales. There are the Scandinavians out in the Hebrides!~ (MacBeth's wife, Lady MacBeth, was the daughter of the Scandinavian King of the Hebrides. In that context, she was probably as normal as any Thor worshiper ever got.)
Then, there are the Highland Scots! These magnificent people are direct descendants of the Irish who invaded and conquered Alba and changed its name to Scota in the 900s.
Most Scots and Scottish descendants are descendants of one or more, or maybe all, of these groups.
The people in Northern Ireland are either Irish or they are descended from a mix of peoples who came from Great Britain to colonize Ulster. They are not all Scottish in origin.
That's why it would be in error to inform the Scots-Irish and the Irish in Ulster that they are the same folks - they are not the same folks!
Are the Picts and Highlanders both Celtic, albeit with different geographic histories? I always assumed the Scots were Celt, but I take your point that the Anglo Saxons probably grabbed the best land. How do the Normans fit into all this? Are they also Anglo Saxon, or some other group? The Normans I know are also Germanic. I am assuming that the Anglo Saxons invaded earlier, per the Arthurian legends, but I am getting wobbly here.