While we never quite get a straightforward confirmation of what you call the "Britain-took-over-Europe explanation," there are strong indications that France had largely embraced British culture by the 24th Century.
The TNG episode Code of Honor reveals that French is mostly a dead language:
DATA: It is a highly structured society in which people live by strict
codes of honor. For example, what Lutan has done is similar to what
certain American Indians once did, called "counting coup." That's from
an obscure language called French --
PICARD: A language which for centuries on Earth represented
civilization, Mister Data.
That's very telling. Anyone who knows anything about the French can attest to the feverishness with which they promote their language and try to protect it from "foreign influence."
Picard is obviously prickly about the perceived slight on French, but presumably Data would have never made that comment if he was speaking it all the time. All indications are that what the viewer hears is his natural voice: he speaks English, with a British accent.
Even if we suppose France itself isn't culturally British, Picard himself seems to embrace Britishness. He alludes to the Battle of Trafalgar before their engagement of the Borg, and views the battle from the point of the view of Admiral Nelson... even though they defeated the French. He knows British naval drinking songs by heart, and is a fan of Gilbert and Sullivan operas. He is enamored with Shakespeare. And yes, he drinks Earl Gray tea.
The sole extent of his "Frenchness" appears to be that he was raised on a vineyard. And while his dialogue with his brother indicates that he was proud of his family's traditions, he never really feels a need to uphold them.
LOUIS: Never did I know anyone less interested in grapes than you, Jean-Luc.
PICARD: No, not true. I was interested. And I was proud that my family were helping to preserve the traditions. I just didn't feel bound by those traditions.
An earlier scene likewise indicates that interest in genuine wine and cooking is dying in France (!?). Picard's brother (and father) are depicted as exceptions to the norm. There's a rather pointed reference to the "danger of losing your values."
(They all taste their red wine) PICARD: Is this the forty six?
ROBERT: Forty seven. You've been drinking too much of that artificial
stuff. What do you call it? Synthehol? It's spoiled you. Ruined your
palate.
PICARD: On the contrary. I think that synthehol heightens one's
appreciation for the genuine article...
PICARD: Leave it to Robert to find the best cook in France, then marry
her.
ROBERT: Yes, but sadly cooking is becoming a lost art. That's your
wretched technology again.
MARIE: Robert and I have had more than a few discussions about getting
a replicator in the house.
PICARD: I remember the same discussions between mother and father.
ROBERT: Father understood better than anybody else the danger of
losing those values which we hold most precious.
PICARD: I don't see that you have to lose anything just by adding a
convenience.
And really, is there anything more British than endorsing rubbish, convenient, bland food? (I kid, I kid)
So the answer does appear to be that to the extent that French culture still persists, Picard doesn't feel the need to stay within its confines and neither do most of his countrymen. That apparently means sounding like a 20th century Shakespearean-trained English actor and drinking a lot of tea.
