I'm not sure what you have in mind for 4e - some (not all) power blocks have a source keyword, but many class abilities don't. Overall, power source doesn't seem to play a big mechanical role other than mediating the interaction of some feats and other abilities that deal with enhancing or recharging. I certainly don't think we're obliged to think of a STR cleric or paladin as fighting very differently from a fighter, even though one is making Divine and the other Martial weapon attacks.I think this misses the point that these categories have actual rules, limitations, etc. attached to them (admittedly minimal in 4e but that doesn't have to be the case in 5e) in the game and thus I think it is pretty important that the fighter's power source be defined in a mechanical sense... just like other classes powers are. Why shouldn't my "demigod" fighter get the benefits and limitations of the divine and martial power sources... while my high level "enlightened" fighter might have the psionic and martial power sources? These mechanical differences give weight to the fiction.
But if, in D&Dnext, these things are mechanically more important, than by all means stick labels on the fighter's abilities, or let the player choose the label s/he prefers. I don't quite see how this would be the sticking point for more mythic fighter abilities, unless one (a) requires fighters to be purely martial, and (b) denies that martial can ever encompass mythic.