That thought occurred to me while I was writing the post. I kind of think that English mixed more than the Irish. In the US there is a tradition of Irish towns, Italian towns, and German towns in cities where the immigrants from the various locales clustered together. I think that kept the cultures of the old country alive and caused generations of people to keep marrying into their traditional gene pool. That way they could easily still say "I'm an Italian American" four generations later. WWI and II may have helped that or hindered it I'm not sure.
The descendants of the English didn't really have that and so mixed in with the rest. I know one elderly gent that can boast of being 1/4 English.