Henry III would certainly not be my pick. His mother was French, he married a fully French wife and he spoke French and the normal language of his court was French.
The first glimmerings of an English king was Edward IV who (horror of horrors) married an English woman (Elizabeth Woodville). Up to this time virtually every single queen was born and raised French. Marrying an "ordinary" English woman was a huge scandal and it alienated him from the court which was mostly composed of French-speaking nobles many with French wives. Edward IV was born in France, but having an English wife who turned out to be a very successful woman, having 10 children, introduced an English component to the court for the first time.
The first kings to be more English than French were the Tudors starting with Henry VII. The Tudors married real English women, not French-bred women imported from the continent. They also started passing laws requiring people to speak English. One of the wierd side effects of the court speaking French was that non-English languages like Scottish, Welsh, Irish and Cornish flourished. After all, you can hardly require people to speak English when the whole court is speaking French! The Tudors changed all this. They made English the language of court and they also began requiring everyone in the kingdom to speak English. The law courts and universities also switched to English under the Tudors. There was still a lot of lingering French among the nobles, but the tide had turned and English became the standard.
This is what one scholar wrote:
Although early Tudor policy affirmed English as the land's primary
language when Henry VII in the early 1490s unexpectedly replaced
statutes published in parallel French and English with statutes
published only in English, this signaled to the nation that the arcane
Anglo-French terminology of law would henceforth be transferred
wholesale into English.
"Studies in the History of the English Language" by Christoper Cain.
Thus you can see 1490 was really the watershed year when Henry VII basically made it clear: ok, everybody, we are all going English.