I think I'd be happy with narrative character advancement - maybe allowing players to adjust or add to their character description as they <narratively> put in the effort to improve.
Of course, I'm assuming that Wits and Grit Dice Rolls are for skilled actions that fall under the purview of a character's description, and Screw It Dice Rolls are more for unskilled actions. (Is this correct?)
For example, this character:
"[Trevor] is a [law-abiding] [District Attorney], who used to be a [Boy Scout]. Always ready to [debate the finer points of law], [he] [encourages] the party members [to live up to his high standards]."
. . . should probably know nothing about how to pick a lock, and if he were in a desperate situation that required him to get through a locked door, "pick the lock" is probably not something that the GM should allow the player to roll for - even using Luck via his Screw It attribute would be strange. ("Break the lock" would be more in-narrative, though.)
Then, if [Trevor] survives, the player might narrate how [Trevor] starts taking some locksmithing courses at night school, adding that information to the character description. (Of course, in-game time will need to pass.)