This video? I think you may have read into the title a little too much (though to be fair, Tom Scott does make… interesting choices with his titles sometimes, relative to the actual content of the video). The point of the video wasn’t to debunk any theories about Shakespeare’s nationality, it was just to compare the linguistic structures of French and Shakespearean/English poetry. Tom Scott does a lot of linguistics videos (that playlist is called “Language Files” after all), so he was merely (yet also provocatively) using Shakespeare’s nationality as a catalyst for comparison between the languages.

As far as I’m aware, there aren’t any serious suggestions that Shakespeare was a Frenchman. I’ve certainly never encountered that—other than hat BBC article, which of course was a joke to make English people shit their pants at the idea that their most cherished writer was actually from somewhere else—so if anyone does earnestly think it, they are on the fringes of the fringe. There also don’t seem to be any French people proposed to be the “true” author of his works, so even those who (frivolously) think someone other than Shakespeare of Stratford wrote Shakespeare don’t seem to think that.

William Shakespeare was in fact born in England, and both his parents’ families had been living there for centuries, so by all reasonable accounts (and maybe even some non-reasonable ones), yes he was English and not French.