First, a clarification: as many scholars have noted, "happiness" is a misleading translation of the Greeek "eudaimonia," and partly for the very reason that is at the heart of your question. The Greeks certainly had disputes over how precisely we should understand eudaimonia, but all would agree that it is the term to use in describing someone who has a good and enviable life (and by thuis, they did not mean good or enviable in a restricted moral sense, but in the very pragmatic sense that all of us would prefer, at least all other things equal). So eudaimonia will be the condition that is always, utterly, and flawlessly in our interest.
Now, this makes what is wrong with a subjective conception of eudaimonia fairly obvious, but since you asked, I will quote myself (and my co-author, T. C. Brickhouse) here:
"Giddy morons may suppose they pursue their interest by doing what only makes them giddier and more foolish, but sensible evaluation will conclude that such lives are nothing to envy. The addict's high, even secured by a lifetime supply of intoxicants, is no model of surpassing success in the pursuit of self-interest. One may be subjectively and even exclusively interested in what is not really in one's interest. For what is really in one's self-interest, one's own personal opinion of what self-interest consists in is hardly decisive. It may be that a certain degree of subjective satisfaction is required for a truly good life. But what qualifies as an authentic self-interest for a given agent [...] is an objective fact about that agent." (Socratic Moral Psychology, Cambridge University Press, 2010, pp. 46-7; emphasis in original)
There is one more thing to add here. As a matter of fact, it seems that people can actually also be fairly poor predictors of what will, as a matter of fact, actually produce even subjective enjoyment. So, one recent study (reported in my daily paper, The Oregonian) sampled 632 Americans, who were asked to rate their own happiness on a scale of 1 to 5, with 5 being the happiest. They were also asked to estimate how much of their spending went to paying bills, buying gifts for themselves, buying gifts for others, and giving to charity. A pattern emerged in their replies: those who engaged more in "prosocial spending" (the latter two categories) rated themselves happier than those who did less "prosocial spending." Now a quote from that same article in the Oregonian:
"A separate study published in 2006 in the Proceedings of the National Academy of Sciences found that the same parts of the brain that produce a good feeling when a person receives a reward also respond when they give to someone else. In fact, the reward areas were more active when giving a gift than when receiving one."
I think these studies (and there have been many others that point in the same direction) show that the very selfish kind of subjective "happiness" seeking that you talk about in your question is actually quite poorly suited even to secude a subjectively "happy" existence. My suspicion is that (for healthy human beings, at any rate), the best way to succeed at securing subjective happiness is to try to do well in achiveing objective flourishing (eudaimonia).
Long story short, I agree with you, and it looks to me as if there is increasing scientific basis for your suspicion, in addition to a very long tradition of philosophical thought in its favor.