…why such a very British-seeming guy has such a flamboyantly Gallic name as Jean-Luc Picard?
(Forgive me for such a remedial question; I’ve watched plenty of episodes, and enjoy it, but was never a huge fan.)
On screen? No. I have no idea whether or not they address the issue in the novels.
My personal theory is that somewhere in the near future, the Second Hundred Years War happens, and this time the right side wins.
is it really so inconceivable to you that someone could learn to speak a foreign language without an accent?
In the absolutely appalling 1st season episode Code of Honor it’s mentioned that French is an ‘obscure’ language, in later episodes he is shown to speak it. He is French, not English.
Off-camera the reason he doesn’t sound French… I recall Steward saying that trying a French accent sounded like Inspector Clouseau was aboard the Enterprise so binned the idea.
IDK how to verify it but I read that Picard would agree to do a character of french origin but wouldn’t agree to speak french or have the character adopt french mannerisms and customs
Of course not, but he never spoke French onscreen, at least that I heard. Given that, I’m left to apply Occam’s Razor to two possible reasons: that he’s an English guy with a French name; or he’s a French guy who for some reason speaks only (perfect) English. The former had been my assumption for the last 20 years. 
Now I know that I was wrong. Picard was a French guy who spoke English…because French is a dead language in the future. 
Picard even visits his brother at their family farm in France. But in “Generations,” his fantasies involve his spending an “English Christmas” with his family.
I think it’s possible for him to be both English and French. It’s not unthinkable in our world today.
Now there’s an idea - a clumsy, overconfident, charming chap in charge of a spaceship. 
Captain Clouseau “Security - there iz a berm in my rerm!”
Security chap “A berm in your rerm?”
Diplomatic Envoy “Captain, does your canine companion bite?”
Captain Clouseau “Non, naturellement.”
Diplomatic Envoy tries to pat dog and gets bitten. “Ow - Captain, you said it did not bite!”
Captain Clouseau “Zat is not my canine companion.”
He is shown to speak fluent French, although so rarely I can’t find a clip but I can point you to the episode; 11001001 he speaks French with a hologram…“Oui, au fond, nous sommes tous Parisiens.”
He was heard singing “Auprès de Ma Blonde” to himself in a couple of episodes, and once reacted to a problem on the Enterprise by muttering “Merde!”
That bit was comedy. French is NOT an obscure language in Picard’s time. Data is the only one not to realise this.
My own headcannon, Picard is speaking French all the time, but the Universal Translator renders in into perfect, accentless English.
Quite so, here’s another Data moment, while Picard admires the tricolor Data starts reciting other flags like the British…“That’s enough Data!”
That bit was comedy. French is NOT an obscure language in Picard’s time. Data is the only one not to realise this.
My own headcannon, Picard is speaking French all the time, but the Universal Translator renders it into perfect, accentless English.
I thought he learned Federation Basic, or English, from an British person.
Consider all the people in Africa or the Middle East who speak with an British accent.
I once met a French lady who spoke with a definite British accent. Same story: She learned the language from Brits, and learned well.
And changes his mouth movements to match. 
He drinks Earl Grey tea just like a proper French gentleman.
I would really like to take Earl Grey into an alley with a baseball bat and discuss with him what tea is supposed to taste like.
But I digress.
He’s French, but just like Klingons, Cardassians, Vulcans and Romulans, he speaks English, because English is the language of space.
He’s a complicated man. But no one understands him but his woman.
Maybe the proper English gentleman who taught him to speak English also introduced him to the charms of Earl Grey?