Addendum: I think a lot of the problem with today's society is that we want the freedom of independence but not the hardships of it, and the benefits of interdependence but not the responsibilies of it.
I think it's worth remembering that true "enlightened self-interest" involves a great deal of behavior we consider altruistic and cooperative. Individual needs are fulfilled much more easily when we have everyone working together to accomplish and/or solve something, with everyone contributing what they can best give, be it skill, time, and/or resources. And, too, everyone prospers when we don't hoard more resources than we need, being willing to share what we don't need, making it easier for you as an individual to meet your needs than if you went it entirely alone. Because whatever benefit you gain in hoarding your own resources would be countered by making it harder to obtain resources you don't have because everyone else is hoarding all of them.
Since the flipside of rendering help and personal effort when others need it, is knowing you in turn will receive help from the community if you ever need it.
Giving up some independence for interdependence is why we get to have the benefits of shiny civilizations versus having to be rural subsistence farmers having to grow and making everything yourself all day. Isn't it a lot easier to get to go to work, do what you're actually good at and enjoy, get some money, and then just go buy the stuff you need from other people who are doing what they're good at and enjoy, rather than having to do it all yourself?
interdependence breeds dictatorships, authority, and malevolence.
Once many/most individuals in society are interdependent, controlling, or being able to disrupt what they depend on others is a great way to control them.
Humans are social animals, and will, naturally group together.
If individuals are independent, these groups will be ground up, and there will be little incentive for anyone to attempt to seize power of a group, as the groups could easily disband, and form anew.
If individuals are dependant on a larger group, someone seizing power over the group, would seize power over all the individuals of the group, and there is little they can do about it. There is great incentive for individuals to seize power over such a group.
I feel a diverse group is always stronger, better, etc than any one can be alone. This can be naturally observed in rainforests, where the soil is very poor but life of all kinds thrives because what one takes out another puts in. On the other hand, a group of individuals that can each do well alone is also much stronger than a group of individuals that are mostly dependent on each other.
Interdependence is rather implied by the fact of being a society, though. In some ways this question reads, do you want to be an island or part of a society?
It's typically easier to help others if you yourself are independent. It cuts out much of the "red tape" between supply/service and demand if you don't have to go asking someone else's permission to do something for a person or group in need. Collective resources usually require collective decision making when it comes to distribution, and that always takes more time than when only one person needs to choose what to do.
And of course, this type of thinking can be scaled up to private businesses or non-profit groups, etc... Anything that manages its own resources has an easier time deciding how to use them than something that has to request those resources from an outside party with its own agendas.
From what I've learned in Psychology, people are naturally social beings, and often live in interdependence. Like where each person is an expert at something, and people go to that person for what they need, rather than one person trying to rely on themselves for everything. Though I like the idea of people being able to be self-sufficient (or at least as much so as possible), we wouldn't have gone very far working alone. Being in groups has its own share of problems too, but we have to take what we can get
Independence and interdependence are not mutually exclusive, nor are they dependent upon established law. In my opinion a government should be based around independence while the state of interdependence should be dependent upon social structure. Nobody is stopping you from volunteering, owning goods/property/wealth in common or establishing/partaking in co-operative business/housing structures in an independent society.
"For example, fixing poverty, gangs, drugs, etc."
Contrarily the more we see society becoming dependent upon corporate monopolies and the law of government representatives, the wider we see the divide between the working class and the upper crust grow, the more motivation we see for the rise of gangs and other criminal groups... drug usage rates increase, the 'War on Drugs' becomes increasingly irrelevant and self-harming, etc.
You are only responsible for yourself so in that sense independence. You cannot be held responsible for the actions of others except under certain circumstances (eg a commander is responsible for the actions of their soldiers etc). However it's also important to recognise that we are social animals and we live in a society where interdependence is vital. Everyone relies on the services of other people to survive. I don't think people have a duty to take care of anyone but themselves but I do feel that it is beneficial to do so.