Obvious objection: Many people desire things other than happiness, and desire them in and of themselves. Ex: virtue, and money. (IV-6/7). Mill’s reply: of course people desire and value many things in and of themselves, irrespective of the fact that those things make them happy. Thus: virtue, money, music, poetry (cf the solution to his depression.) However, those things are still connected to happiness, via an associationist theory of happiness: it is true that agents desire things for their own sake rather than as means to happiness. However, those things have become, in virtue of their habitual association with happiness, part of happiness. Accordingly, when we desire them, we desire them as part of our happiness.
Even if they are part of happiness, the point remain that we do not necessarily desire them because they are part of happiness – and yes this is what claim 3 says we do.
Many of the things which we desire, and which Mill claims are part of happiness, in fact cause us no pleasure at all: cf the example of the parent who would rather learn thathis child is dead, than continue to live with the uncertainty of not knowing whether he is dead. But if happiness is defined as consisting in pleasurable mental states (which is what Mill believes), then it is hard to see how knowledge of that kind could be part of one’s happiness.