And another bloody thing, I've yet to meet anyone who considers themselves principally to be British. I don't, I'm English, not Welsh or Scottish, nor a blend of all three.
In my opinion, you are making the same mistake that an Israeli of Ashkenazi, or Sephardic, or native origin (Sabra) makes, when the entire world thinks of Israelis as "just Jews." So, since many Americans, I believe, do not know where Wales is on the map, they just lump everyone from the British Isles into the moniker British.
Only Sherlock Holmes and Watson are known to be English.
By the way, where did I hear that the Royals are of German ancestry? The name Windsor used to be some German name?