Is there a reason, other than safety, to cook the egg mixture when making a custard-style ice cream?
Do you mean to cook the custard? Obviously, food safety, as you mention. But you also want to coagulate some of those long protein strands because that is what is going to give you that lovely, tender consistency when frozen. If you let the cooled custard sit in the fridge overnight, they get even fatter and the result is even more wonderful.
As boulangere said, it's not a custard unless you cook it.