I think it's a bit odd that Mill uses this as a basis for a moral. I feel that it should be understood already that people will do whatever necessary to attain happiness but why would it automatically become a moral to abide by. He claims, "The sole evidence that it is possible to produce that anything is desirable, is that people do actually desire it". This theory is easy to believe but with a little "background check" is quite a basic derivation of natural human instinct. He then tries to apply the argument to society as a whole, saying that if an individual desires what would cause him the most individual pleasure, then all of society ought to desire that which would, on balance, add the most pleasure to society. The leap of logic from individual egoism to group egoism is again, difficult to justify, but easy to believe. According to utilitarianism, everyone would have to make the same decision when placed in the same circumstances, because everyone would have the same ends: the aggregate good. What this means is that in a utilitarian society, while we would all have to consider the good of others, we could be confident in the fact that they were also considering our good.
Mill is able to reduce morality to scientific fact. Certain things are right, and others wrong, according to some quantifiable measure of "aggregate happiness." He asserts that science will someday find a way to measure the level of pleasure or pain that a given situation causes a given person, and that there is some way to compare those measurements for all of society to determine the right action.
As much as the idea of only having to ask whether something will create happiness without having to apply it to some outside moral standard which may provide an uncomfortable answer is attractive, people are suspicious of a theory which makes being moral as easy as counting happiness units. As well, they worry that some of the things that make people happy are not worthy of the title "moral.