Mill's argument for Utilitarianism

As I understand it, Mill's argument for utilitarianism runs something like this:
1) Pleasure is the only thing desired.
So: 2) Pleasure is the only thing desirable.

Then critics point out that this can't be right: crudely, assuming the first premise is true, why can't people just be desiring the wrong things?

But isn't there a similar but better argument?

1) Everyone desires only pleasure.
2) People aren't all that stupid.
So: 3) Pleasure is the only thing desirable.

This is still crude, but my aim isn't to exonerate Mill entirely. All I wish to point out is that there must be a decent argument lurking in the area which appeals to the fact that everyone has "the intuition" that pleasure is desirable, and that constitutes evidence that pleasure is, actually, desirable.