This is a response to someone asking about the purpose, role, or usefulness of the cognitive function Introverted Feeling (Fi) in the larger picture of society and helping others solve problems, not just it's importance within an individual. This is about ways that FPs (ENFP, INFP, ESFP, ISFP) contribute to the lives of those around them.
Fi sifts through all the priorities and obligations and hones it all down to what is truly important, reminding people of where the real value and joy in life is to be found. It reminds people to notice and enjoy beauty, which can help renew people with a sense of joy and peace.
Fi's insight into the workings of the internal self can be quite valuable for personal emotional/psychological/relational growth, not just to the individual with Fi, but to other people as well. Remember that Fi is working with information from the perception functions Extroverted Intuition (Ne) or Extroverted Sensing (Se), so it's not totally blind to what's going on outside itself with other people. Fi's recognition of it's own individuality helps it to be aware of and accepting of Different ways of thinking and feeling within other individuals, allowing it to provide understanding and insight about others who are not the same as themselves. Fi's ability to 'see' the inner 'essence' of another person prompts compassion and gives the Fi user a very personal touch when encouraging, comforting, or inspiring someone. Fi's deep understanding of emotions can also 'touch' others indirectly when they 'connect' with Fi inspired art and personal expressions.
I think Fi's value of individuality doesn't end with itself, but values the individuality of everyone, often prompting someone to act as the champion of minorities, of people who are overlooked or don't fit, of alternative movements, different ideas, and new styles of art, etc. I think Introverted Thinking (Ti) does some of this too of course, but both types of Introverted Judging (Ji) have their own approach which are useful for different people and situations. I believe Fi values diversity and helps prevent homogenization. While Unity and Harmony are good, and may also be valued by Fi folk, they can also become static and oppressive. Variety is the spice of life as they say, and while I'm sure there would still be variety without the FPs in the world... I suspect there would be less of it.
I think one of the important roles of Fi is standing up to group values when they start running awry, especially when it comes to remembering the needs and interests of 'the few'. While Fi dom/aux people aren't the only ones who can do this, Fi seems most likely to question values that are culturally/religiously taken for granted, to motivate someone to instigate a change of values, and take to the brave step of being the first to stand alone against the crowd (Ti probably does this more with concepts and systems, Fi more with morals, matters of compassion, and style).
Regarding the question about solving problems:
Determining your true priorities, and gaining understanding of your internal emotional/mental landscape helps to give a person direction, helps give them insight into what the most sensible course of action will be for them as an individual with the overall personal results in mind, not just the current specific situation. Gaining insight and appreciation for how other individuals feel and value and think about things can help tremendously with relationships. And it's amazingly empowering to have your individuality appreciated.
Regarding the misconception that Fi is 'selfish':
There is no selfish function. People are selfish. Despite the stereotypes out there, I don't believe there is any type that is more likely to be selfish than others.
Introverted Feeling takes the observations from Ne-Si or Se-Ni and internally evaluates what is most important to that person as an individual - this includes factoring in the values of others that it is aware of. In my experience any moderately mature FP is well aware of the subjective nature of their evaluations and has great respect for each individual person's ability and right to determine what they believe is important for themselves, which leads them to be accepting and non-pushy in their interactions. When Fi has determined that the needs of someone else are more important than one's own personal desires or needs, it results in selfless, generous, compassionate, helpful attitudes and actions.
Fi does tend to resist the will of others when it does not agree, but at least when mature it also allows others to disagree and go their own way as well. While in SOME cases resistance of other's will may indeed be selfish, it is my opinion that far too often people are quick to accuse someone else of being 'selfish' just because they are themselves selfishly annoyed that the other person has not complied with their own will. This accusation implies that other people are not allowed to make their own choices or stand up for themselves - lest they be shamed for being 'selfish'. Non-compliance is not always the same as selfishly caring for no one but your own interests, and it is not inherently laudable to always acquiesce to the will of others.