I agree that it adds an incentive, but I don't think it's literally the only way to make them profitable. And not every city necessarily cares that they are profitable.
In Northern VA they increased the yellow light time on intersections with red light cameras specifically because they only want to catch "genuine" red light runners.
Not only does it create the incentive to shorten the red light, it's practically the only way to make them profitable for cities in many places.