Surface contact is one reason you want an ice/water slurry instead of just ice, but the real reason is that ice melting consume a lot more energy than just ice being warmed up to it's melting point.

The ice will quickly come up to it's melting (equilibrium!) point, without cooling the ice cream mixture very much. Remember, we're trying to freeze the ice cream (not just cool it down), which is proportionally just as thermodynamically expensive as melting ice. Bringing the ice up to it's melting point alone won't suck enough heat out of the ice cream mixture to freeze it.