D&D isn't a game about the everyman it's a game about Heroes with a capital H. The back of the Player's Handbook proudly proclaims, "Create Heroic Characters." It's a game where you adopt a fantasy persona such as "a mighty Fighter, a cunning Rogue, a faithful Cleric, a magical Wizard, or something else." If you want to play an everyman with humble origins, that's what backgrounds are for. Yeah, you started out as a baker, but now you're a pious Cleric battling the fiendish machinations of Iuz.
There were Kits from AD&D 2nd edition you could give classes to reflect a modest origin. Peasant Wizard and there was one in the Fighter's Handbook called Common/Peasant Hero or something like that. I think it's a fine idea for a background, but I'm not sure what purpose an everyman class would serve. Is it a martial class? Magic?