Just like with Chekhov, I believe the intention is to show that Starfleet isn't just a "United States" space armada, but is for the entire Earth (and maybe even the Federation).
The Next Generation Bible (Writer/Director's Guide) states this:
Born in Paris, France, Picard betrays a gallic accent only when deep emotions are triggered. Otherwise, since ethnic accents are no longer common, he carries only a touch of French phrasing in his speech.
Obviously the onscreen version of Picard had a much more pronounced English accent than French.
Memory Alpha has this to say:
The character of Picard changed quite a bit from the original March, 1987 Writers/Directors guide. In that guide, he was born in Paris, carried a touch of French phrasing in his speech, and pretended that France is "the only true civilization" on Earth (reminiscent of Pavel Chekov). The latter only surfaced briefly in some early episodes (most notably "The Last Outpost"). Over the course of the series, his character became more English. He enjoyed the works of William Shakespeare, and was never seen reading a French author. He drinks Earl Grey tea, an English beverage named after an English nobleman, and was rarely seen drinking wine (he only drinks it on three occasions, once in "First Contact", again in "Family", and once at the end of Star Trek Nemesis), a beverage which is commonplace in French life. Additionally, his brother and nephew both spoke with English accents.
Memory Alpha goes on to say that there were a number of famous explorers / scientists named Piccard / Picard.
It is also worth noting that a "Picard" is a sixteenth and seventeenth century term for someone of an enquiring mindset (derived from the birthplace of John Calvin of Picardy or Picardie). With his interest in the development of Humanism, it is likely Roddenberry knew this. Certainly it is a fitting description of Jean-Luc's character as well as his name.
For the rest of the human bridge crew:
Riker is the only regular bridge crew member who was born in the United States.