^I'm guessing violence in this case means more general action than gore. In that case there'd a lot more to eliminate from racing than just Carmageddon, as you'd have kart racers(the various weapons you attack the other racers with), the Pro-Am games, Rock 'n Roll Racing, Death Rally(which may have gore, been a while since I played it), The Chase HQ series(where your objective is to ram a criminals car until it breaks down and you can arrest them), obviously Road Rash, even though it's cartoony(like a Kunio game), there was a similar Super Famicom game to the concept of Road Rash, etc.
That said, for a driving game, I don't think Crazy Taxi has been mentioned. People will dive out of the way if you're about to hit them, so no contact, and obviously no gore or anything.
The Super Monkey Ball series is effectively a spiritual successor to Marble Madness. Well the core game anyway. There may be mini games throughout the series that have the monkeys attack each other in some form or another. I don't know if mini games should be enough to DQ a game as non-violent otherwise though.
Otherwise, persoanlly I kinda prefer violence. Beat 'em ups are my favorite genre.