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.
They can be treated very well in certain systems, to the point where they aren't as much of a problem. There are countries that have very little poverty and cultural groups that have almost no income disparity.
Europe as a whole has around twice the population and nearly five times the homeless population of the US. Canada has around 1/10th the population of the US and 1/7th the homeless population. Japan has a nominally low homeless rate, but has a tendency of locking its homeless up in mental institutions, a policy which largely ended in the West after widespread abuses in the mental health institutions were publicly revealed in the 60's and 70's. (Additionally, Japan systematically underreports its homeless population; it refused to even acknowledge their existence until 1999.)
These problems -haven't- been solved. At best they're swept under the rug.
I think the question about a stronger economy is very interesting. It sort of depends on what you mean by strong.
In a country like the US you can get, at times, extremely large growth. Our society highly values independence, as well as self-interest within a certain bounds, and this results in competition, which allows for constant improvement (as well as a lot of people losing along the way, but it could be argued this is healthy).
I have a professor who studies culture's role in an economy and he has found that in Japan people are extremely altruistic and have a culture of community, to the point that they will concertedly avoid making purely self-interested decisions on both small and large scales. I do not know what the result of this is on the economy. A high amount of pride and interdependence can result in less willingness to take risks, perhaps, and lower growth, but also create certain other effects that give it strength or stability. I do not know.
An economy will grow more if there is more individual independence, probably. However, a 'bigger' economy does not necessarily satisfy everyone as being the best indicator of well-being, although it does raise improve many tangible things.
From what I've heard of Japan, people frequently get stress related problems because obedience and honor are so valued. So people overwork themselves putting far too much effort into something and progress more because of the interdependence there. Basically you're honor-bound to your boss, people work harder, possibly harder than people who are out for themselves.
I think that a combination of both is definitely possible.
Some of my relatives are broom makers in the Arkansas Ozark Mountains. They are homesteaders who live off of a driveway that's a mile long on a dirt road that's a decent drive to their nearest neighbor. Yet the culture in the area is very interdependent. People share all of their crafted goods even though they all are completely capable of surviving without the support of others, and if they had self-governance, would be much more likely to try and solve problems as a group rather than putting the blame on anyone in trouble.
tl;dr: move to the Ozarks because it's practically a utopia for people who want to work together but also value independence. Plus it's really pretty and everyone is nice.
There's a concept (not sure if its actually been done before) where a community would set a radius of 10 or 30 miles, something like that? Anyways, no products or services were allowed to enter or leave that radius, so everyone was dependant on everyone else inside that radius. So that means no importing oranges if your community wasn't able to grow oranges, and you wouldn't be able to export potatoes to other places.