Re: "We won the battle, but who won the war (not us)"
"As I note above, neither of the sides in the Hundred Years War were "us"."
Most of the armies' complement would not have been from the knight upper classes. As to lineage - it may be difficult to trace a family tree but some English family names indicate a Norman ancestry. Not sure how well DNA testing can differentiate between the English Anglo-Saxon/Danish stock and the French Norman stock. Basically the same invaders settled in both places.