This is exactly the point I was about to make.
I've always found the Celtic emphasis highly exaggerated (considering the majority of my ancestry leads back to those same Scots-Irish groups).
As you point out I also find it dubious to consider the Scots-Irish highly "Celtic". To clarify we would be referring to the insular Celts vs Anglo-Saxon. Being on those border region they were not only culturally infused with English culture and more removed from Insular Celtic culture, additionally they would be genetically far more diverse and mixed than what we would classify as "Celt" (Insular Celt).
On way to look at this is DNA, here are Y DNA haplogroups of modern tested individuals. Though this doesn't represent the area 300-400 years ago it does give us an idea of the population groups and still gives us a nice parallel to the known history.
A few genetic maps.
A good overview
http://www.eupedia.com/genetics/britain_ireland_dna.shtml#maps
Some more in-depth ones for the groups
R1b - L21
http://www.eupedia.com/europe/Haplogroup_R1b_Y-DNA.shtml#L21
R1b - S21/U106
http://www.eupedia.com/europe/Haplogroup_R1b_Y-DNA.shtml#S21-U106
R1b - S28/U152
http://www.eupedia.com/europe/Haplogroup_R1b_Y-DNA.shtml#S28-U152
I1
http://www.eupedia.com/europe/Haplogroup_I1_Y-DNA.shtml#distribution
The summary the DNA shows matches the history. R1b L21 is called the "Atlantic Celtic" Haplogroup. It originated about 4400 years ago and likely arrived in Britain around 2100 BC or so.
This makes it the most common haplogroup of all the British Celtic societies, mostly surviving in hotspots these days in the Insular Celtic groups, though we should note that some Y DNA lines survive from the other British Celtic groups that happened to persist despite other populations that came.
Lowland Scotland is made up of a good portion of this, though again remember a portion of that is not just the modern Scottish Celtic culture but that which did infuse and adapt into English culture as well (though to a lesser extent). Also L21 represents some Continental Celtic groups too, not just the British Isles.
R1b - S21/U106 and I1 represents both Anglo-Saxon and Viking movement into Britain, most of it likely from Anglo-Saxons. You can see the map really maps the historical record in this case.
R1b - S28/U152 shows a later Celtic migration (a cousin branch to L21 though not the same) to Britain. It hotspots in Northern Italy so follows well the movements of La Tene and Halstat Continental Celtic groups. This happens to be my Y DNA group that likely went to England. They came later and made the smallest impact. Also this group likely came in additional waves with the Romans (Italy being a good portion U152 so the Romans likely were too) and of course the Normans or other migrations from France/Belgium over to Britain.
With all this considered you can see that not only was lowland scotland a cultural hodgepodge between the Celts and the Anglo-Saxons (which still contained trace elements of Celtic Britain both in culture and DNA) but DNA wise they were too.
All of this makes me suspect the Scots-Irish weren't ultra Celtic like people think of these days... More Anglo-Saxon/Celtic that the lowland Scotland and border English region would suggest.