About Picard, I always took it as him being schooled in England as a young boy. I've met a number of folks from India who were schooled in England and most have a fairly prominent English accent. What would have been great in TNG is seeing Picard throw out a few French phrases here and there... which would have help bolster his French background. Or even running into a French friend in Starfleet, where they'd start off speaking a bit of French before shifting over to English.
As I recall, the Picard character did lay some French stuff on us very early in TNG, referring to a fellow ship captain as "mon ami" and also waxing nostalgic about the order of the colors blue, white, and red in their appearance on the French flag.