I don't think they had any interest any creating a conventional "actual" person like some bog standard Hollywood drama - all along they said it was basically a fairytale, mythic, heightened reality. All you need to know is he's a violent man that does violent things - in the name of love/justice. If there had been some cheesy scene where he revealed some backstory to her it would've been too phony. Imagine if Clint Eastwood's Man with No Name had delivered an impassioned monologue about how his family had died horribly and he was looking for revenge, the mystique would be ruined.
There's no doubt Gosling can deliver the sort of performance you're talking about (choose pretty much any of his past movies) - but they were going for something very specific with this film. And I don't think his lack of dialogue meant his performance lacked depth, nor do I think the scenes of violence (of which there are actually only 3 explicit ones - the hotel room scene, the elevator scene, and Cranston's death) meant an abandonment/devolvement of the plot or character at all. They were integral to the piece as a whole. Just watch Gosling in the hammer/bullet scene: when talking on the phone to Nino he repeats the same lines he has used before, but see how Gosling lets the Driver's adrenaline and emotion kick in here, the cool demeanour he exhibited previously struggles to assert itself in the face of the actions he carries out. Or take the diner scene where the guy approaches him and he tells to leave him alone or he'll shut his mouth for him - just watch his reaction following that outburst. Or yet another example, that long slow shot in the hotel room as he moves backwards out of frame, his eyes wide, processing the shock of gunfight. That's all character.