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I'm not at all sure that Hare's argument works.

I agree that there is a sense in which moral judgments must be universalizable. (What is right for A to do in circumstances C must be right for B to do under the same circumstances.) And I agree that morality requires impartiality in some important respects. (I would be morally wrong to say that I value only the good of white people and not black people.) Those principles might allow us to conclude that we must consider the good of all persons equally when we make moral judgments.

But classic utilitarians want to take an additional step. They want to say that we should maximize the aggregate good that we produce. That's the sticky wicket.

I claim that this last step cannot be justified even by a combination of universalizability and impartiality. I will go even further. I claim that by taking that last step utilitarians undermine one of their own premises; namely, that everyone's welfare is to be valued equally. That's because once we start maximizing the aggregate good of some sort (satisfied rational preferences, for example) we end up trading off the good of some people for the greater good of others.