Nah, there's already been plenty of buildup to suggest there's a more heavy meaning behind that intial.
This is probably one of the most well known examples:
But there's also a few smaller examples, such as this:
View attachment 307813
Where people who have that initial, are treated as different from the more common folk, and apparently from experience too.
Another thing I feel I should point out, is the fact that all of the carriers of D, at least, for now, seemingly aren't related by blood.(Whitebeard and Luffy obviously aren't long lost cousins lol) A middle name is a family initial, so why would so many seemingly unrelated characters have it if they weren't connected in some way?