Our impression of what stock we come from is likely strongly influenced by what genealogical lines we know something about, rather than the actual numbers. I know parts of my two grandfathers' lines back 1) beyond the Mayflower into the 16th C and 2) beyond the Anne back into the 13th C. This tricks me into thinking I am deeply Puritan/East Anglian/Saxon in heritage. But those lines are only a fraction of even the section they are in. One grandfather - so, 25% of my DNA - does indeed seem to come from a network of early Puritan lines - Wymans, Doanes, Richardsons, Spinneys, Crowells, Hopkins, Snows. But the other is mostly Scots-Irish with a thread or two of Puritan Ponds, Smiths, Balls, and Hawes way, way back. Gramps's line is about 75% seige-of-Derry stuff.
One grandmother was Swedish on both sides, so 25% of my DNA contributors 200 years ago were all eking out a precarious living near Lake Vannern in Gotaland.
The remaining 25% - a big chunk, really, as big as the lines I focus on - come from a grandmother born Ruth Irene Neat, who died before I was born. Her lines all seem to be Welsh and North Midlands. That was a whole separate migration, to different parts of Massachusetts. And most of the lines seem to have fizzled out. I and mine may be among the last representatives of a few lines on their last legs.
I don't think of myself as Welsh. We keep no Welsh customs, identify no immigrant ancestors, study no Welsh history. But it may be as strong as the others.