I am glad you find your way here because your post inspired the whole thread.
They exist because they are a consequence of our biology and an upgrade of our natural instincts. However this thread is more about classification than what is what in real life. I have seen plenty of times that people on this forum regard feelings/emotions as rational. What I find problematic for a number of reasons and the main is that if emotions are rational then there is no decision making process left that will be irrational ... and I have yet to see a person that will defend the position that all humans are completely rational. (since it is obvious that they are not)
Therefore I insist that emotions are irrational decision making in both feelers and thinkers. They can provide good results and that is why they are part of human mind, since is questionable if we would survive if there were none of them. However there is a complication and that is called "technological development" that has a very strict rules in what is possible and what is needed, therefore once you become dependant on technology you simply don't have the luxury of following impulses as you like since the consequences will be devastating. However technology provides great increase in efficiency and it is therefore accepted. I have spent decades studing the Earth/environment science and some of it was on official university level, plus geopolitics is my side hobby for quite some time as well. Therfore when you take a look at objective data, formulas and situation in the field it becomes very quckly obvious that humanty as a whole does not know what they are doing. Even if we throw climate change out of the equation the whole system is still pretty much unsustanable for fast growing population.
The reason why I am saying this is because I am trying to show that what looks good on paper such as more people or less poverty can cause much larger problems. However if satisfaction of your own emotions blinds you or prevents you from seeking deeper truth they become observably irrational (what they have been all the time). This may not be so visable in completely down to Earth situations for which emotions are actually made for, therefore other functions and people need to jump in to balance this out. The trick is that just if it feels good and right that does not mean that it is the right choice. (ask fat people for confirmation). From what I have seen most people have never detached enought to see the humanty from a more of a outside point of view and therefore it is easy to mix rational and irrational. Because of this I think that it is important that one decision making process should be irrational because that will result in a model that will much better describe actual human bahavior. While if you have both sides as rational then the whole system is basically loosing it's point.
But to trully answer "What is the purpose of feelings" I would probably need to write a book and I am just not enought of a expert for that.