I would respectfully argue though that while the British nobles at the battle certainly possessed French ancestry and some residual identification with the culture (and were keen on getting their ancestral lands back) the common foot soldiers and archers were as English as the Thames. Their unfortunate opponents did not believe they were fighting fellow Frenchmen and it is celebrated in England as a miraculous English victory. Shakespeare (and Olivier in 1944) certainly played some part in making that so.
PS: I enjoy reading your posts. European history was not my focus in college. Middle East was.
I distrust Shakespeare's version of the events -- it's good literature but bad history. He wrote at a time when the concept of a nation was full-blown while during the Plantagenet wars the concept of the nation was not so distinct.