Perhaps the fact that French was the language of the noble and the learned in England until the first quarter of the Fifteenth Century does point to the peerage and gentry being somewhat more...cosmopolitan in their fashions and preferences, but all the same, we must not forget the shock that the kings and nobles of England faced after the battle of Bouvines in 1214.
From a nobility which held lands in England and France, very much the epitome of the feudalism you describe, they were reduced to merely kings and lords of England and Gascony for quite some time. The king of England, John, was forced, at least for the moment, to merely be King of England and Gascony, and not king of the Angevin Empire his father, Henry the Second, had forged. This had a profound long term effect on the nobility of England, for it forced them to be merely English earls, English dukes and English kings, despite the language they spoke and the culture they held to be best.
And even if they spoke French, they did not comport themselves in exactly the same way as their continental equals. They had their own type of culture, the result of the intermingling of their original and eclectic roots with what they found in England and the long years in which they coexisted with the Anglo-Saxons. These men were different from both their Gascon subjects (which cannot be called French either, on account of many differences in both their language and their culture) and from the nobility of France.
The One Hundred Years War, however, clearly defined the two sides of the Channel as being different. If before the English nobles were similar to the French in many aspects, both during the war and after it, the differences brought about by years of incessant conflict, killing, ransoming and looting eventually led to the English nobility abandoning French as their language and using Middle English. This can be seen at its best in the tales of Geoffrey Chaucher, who wrote in Middle English and was a nobleman at the same time.
And as for Henry the Seventh being Welsh, well, maybe he was Welsh by way of blood but even then, culturally he was more English than any Welshman. He had to be, if we wanted to rule a country of English nobles. His claim to the throne came from his mother, Margaret Beaufort, after all, and not from his father.