Before I can help you try and figure out why Aristotle rejects the contention that happiness is pleasure, honor, or wealth, I think I should make sure both you and I have some idea of what Aristotle means by happiness.
It seems like Aristotle views happiness as a form of “self-sufficiency. Aristotle puts out the idea that happiness is the “final good” since “the final good is thought to be self-sufficient.”
He clarifies what he means by “self-sufficient.” It’s not literally “self-sufficient.” It’s not restricted to one person. It includes a person’s family, friends, and even the other members in their community. It’s as if Aristotle is saying happiness is a kind of harmony. It brings together and connects all of the people in your life.
In fact, Aristotle actually uses the word "harmony." He says, “With those who identify happiness with virtue, or some one virtue, our account is in harmony.”
With pleasure, wealth, and honor, there is neither harmony nor self-sufficiency. As Aristotle writes, people choose those three things “for the sake of happiness.” Yes, people like honor, wealth, and pleasure in themselves and separate from happiness. Yet Aristotle insists that you can’t dismiss happiness from the equation. These are things that people use to try and gain happiness whether they admit it or not.
For pleasure, specifically, Aristotle says that the virtuous and happy person possesses a life that gets “its pleasure in itself.” This person has no need to pursue separate pleasures. They’re self-sufficient (to go back to that word).
Wealth is not—or should not be—self-sufficient. The person entirely dedicated to wealth is neither happy nor virtuous. They’re “vulgar” because they’re “spending beyond what is right.” Again, as with pleasure, wealth is “merely useful and for the sake of something else.”
An inordinate focus on honor will not lead to happiness either. For Aristotle, too much focus on honor and acquiring esteem and respect will produce an “empty vanity.” Conversely, not caring enough about honor will lead to “undue humility.” As with wealth, honor should be pursued with temperance.
It seems like pleasure, wealth, and honor all share a fragmented character. They’re a part of happiness but they’re not happiness in themselves. Perusing one will not produce happiness, it will lead to too much pleasure-seeking, inordinate wealth, or vanity/timidity.
For Aristotle, it seems like happiness is needed to bring pleasure, wealth, and honor together in a harmonious and balanced way. Perhaps that’s why Aristotle calls happiness “the final good.” It’s the last step. The final piece of the puzzle. Without happiness, pleasure, honor, and virtue never become anything more than what they are. They won't acquire a higher meaning.