Sorry, I don't mean to say that it has to be a product. All I'm saying is that the product formulation is one way to achieve a complex-value effect.

A product is the unique way to evaluate a group of values if we want to have the property that whenever we hold all values constant but one, the result scales linearly with the remaining value (or, to avoid the "linearly", we can apply some additional math to the value first). I don't think that in general this is true of our utility functions, but it might sometimes be a useful approximation.