It's a bit of an oversimplification to argue that "Money cannot buy happiness" is what makes Fitzgerald's work a great piece of American Fiction. Certainly, this element is there, but I think that that the theme that is present is a bit more nuanced that such a statement. In my mind, if this were a theme of the work, it would be able to easily criticize the characters in the book and not engage in any self reflection. I think this is probably where the book speaks its loudest. In analyzing the people of West Egg society, the reader is able to engage in self analysis and reflection about the nature of dreams and how we go about in pursuing them. In the end, this might be where the challenge of the work is and the most relevant part of Fitzgerald's writing is one where the reader is able to analyze themselves and their own actions in the light of what is being read.
Concerning The Great Gatsby, you should be careful of looking for simple, easy one-liners in sophisticated fiction. Human existence is complex, not simple, and sophisticated fiction usually reflects that.
For instance, in this novel, money is only a means to an end for Gatsby. He doesn't strive to become wealthy because of greed. He strives to become wealthy for the sake of winning Daisy back. Money only matters to Gatsby because he thinks it will help him win Daisy back. An easy one-liner about money not buying happiness doesn't apply to Gatsby. Daisy will bring happiness to Gatsby, nothing else. That is the issue.
For another example, Tom is definitely leading a happy life, so to speak. He has a beautiful wife, doesn't have to work as far as the reader knows, and has a girlfriend, too. And he wins in the end. He is so sure of himself and so ignorant that he thinks he's always right and enjoys the winning. He has what he wants and gets what he wants. If one wants to apply one-liners to the novel, one could argue that the novel suggests that money can buy happiness. A reader's judgments concerning Tom do not take away from the fact that Tom is happy with his situation and his life.
The corruption of the American Dream is at issue in the novel, as are one's ability or inability to recapture the past, illusion, and other issues. But saying that the novel shows that money can't buy happiness is too moralistic and too easy and too simplistic.