Yes, I figured Doohan was probably pulling our collective leg with that “future Scotsman” remark.
Even so, it's not without merit. Accents really do evolve over time. The Scottish accent of today would probably be unrecognizable to a Scot from 300 years ago, and so it stands to reason that a Scottish accent 300 years from now would be rather unlike the present one. Same with a Russian accent like Chekov's.
People sometimes complain about the Frenchman Jean-Luc Picard having an English accent. My response is, who's to say that isn't what a French accent will sound like 400 years from now?
James Doohan was well known for being able to pull off accents very convincingly. So, it does stand reason to believe he modified the accent to be more intelligible to the general audience (primarily USA at the time). And given how accents do shift in time, it fits in nicely with this hypothesis that it was a "futurized" version of a Scottish accent.
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.