“They were Norman, not French. And they ended up thinking of themselves as English anyway.”
That is not valid. If every time you disagree with something about France you say, well, he is not French, is Corsican (Napoleon) or, Alsatian, or Auvergnat, or Gascon, France doesn’t exist as country. France is composed of various counties and even different languages, before the great languages unification of the XIV century.
It is like saying, well, he is not English, he is Londoner…
The Normans were French.
The First English King buried in England and not in France was John Lackland.
And the reason why during the Hundred Years War, the English King was able to claim the Throne of France was because technically, he was French. And without the superb finding of the so-called Salic law, he would have been (no transmission of the throne to female).