Surely part of the issue is that the ice at < 0C while the liquid portion is at 0C (because of the equilibrium thing) - but it's the liquid portion, not the ice, that's most physically connected to the inner container you're trying to freeze (this is the important point).
If you add ice you reduce the equilibrium temp and as a result the < 0C ice temp can be passed to the liquid phase and as a result on to the inner con tain er where you're making the ice cream
If you add ice you reduce the equilibrium temp and as a result the < 0C ice temp can be passed to the liquid phase and as a result on to the inner con tain er where you're making the ice cream