Since feeling good is so fundamental to getting what we want in life, why isn't it an appetite too?

It is.

The desire for food is an appetite. The desire to feel good (I would call it the desire to expand, or the desire to self-actualize) is also an appetite. There's no fundamental difference between the two, other than priority (survival takes precedence over self-actualization). You can say, well, one is felt in the body and the other is not, but they are both felt by the body. Our nervous system is wired for this.

People think about what they want before they get it. In thinking about what they want, they feel some (or all) of the same feelings about it that they will when they actually get the thing that they want (in other words, they feel good about it). This is the drive that causes us to dream. In the reality-creation world, you would say that they already have the thing that they dream about. But in the physical world, it is the fundamental drive to expand that causes us to have children, build bridges, and compose sonatas.

Why do we have so much leeway in being able to feel bad?

Because the same kind of correction signals that apply to getting food for the body also apply to getting food for the mind. It is these signals that help us achieve and maintain balance.

I think that in the same way we feel the need to eat, drink and have sex, we also feel the need to satisfy our happiness.

I used to argue that not everything we do is selfishly motivated but now from experience, I know that everything we do is selfishly motivated. To me, that implies that we are satisfying our own desire to feel good and be happy. Would you not agree? Therefore, we must also experience the need to feel good, probably a lot more than we think.

Now, I think there is a distinction though. If you think about the way you have been brought up, would you not say that you find it easier to satisfy those needs that you have been told were ok to satisfy since you were born. How many people get told since they are kids that life is supposed to be fun and that we are supposed to do things that make us feel good. I think more the opposite.

So, I think it is not that we do not have the instinct to fulfil that desire to feel good just like we feel the need to fulfil our other needs, it is just that because of our surroundings, we have been trained to believe that it is harder to fulfil the desire to feel good.

Sex : To stop us from forgetting to procreate (and dying out as a species)

Yeah, right!

Why do we have so much leeway in being able to feel bad?

Because every sexual adventure did not result in a procreation?

Sorry, Stingray, but you asked for that one.. hope you have a funny bone.

In a way I think it is, although I'm not sure I'd use the word appetite. Our body will die or cease to be if we fail to satisfy our thirst and hunger, we just experience a different kind of death when we don't heed that yearning to feel good. Our body, mind and spirit all need to have their 'appetites' sated in order for us to have a contented life, so I think this could quantify 'feeling good' as an appetite.

Thirst, hunger, and sex are all lower, more important needs than "feeling good" is; on Maslow's hierarchy of needs.