I like your question, and I really hope that it stirs up conversation into interesting directions.
The immediate problem that comes to my mind with having an anarchist culture is that it requires positive values. By which I mean things such as "mutual aid," "solidarity," "free association," etc. These are Anarchist values, and as such are a response to capitalist values. Often they are the opposite, and the opposite always contains the thing it means to oppose. So if we are to try to adopt our own values in the form of a culture, we are accidentally reproducing capitalism.
It's fashionable among post-left types these days to emphasize that anarchism has to be a negative project if we don't want to fall into this trap. That means that everything we do and value has to be for the sake of negating the present social order. Then, if that happens, we should scrap all of it and let the new social relations dictate values and forms-of-life, in ways that we could never begin to imagine from our current viewpoint. "One must burn the black flag after one burns all the nations flags."
But this train of thought is unsatisfactory to me. For one, most insurrectionary types and nihilists don't believe there will ever be a 'total anarchist triumph!' So holding off on creating a culture until then, and focusing only on negation and destruction seems like a trap that isolates anarchists and our ideas. For one, Anarchist values as stated above are popular. Mutual aid, free association, and autonomy are things that most people are down with. Projects that try to do what you are saying: Really Really Free Markets, Free Schools, and even Food Not Bombs tend to attract people who are dissatisfied with capitalism.
Anarchists usually organize themselves along these values, even if its just in the form of a collective house. Even the nihilists. It's useful for us. We like it.
I think that the only real problem with above-stated projects is that there is a misunderstanding on the part of the people who tend to do them. They think they are prefiguring another world. But that doesn't mean the projects themselves are useless! Anarchists into more fashionable ideas focus on this downfall, while discarding the real potential that these projects have. I think this is carry-over from an insurrectionary purism that rippled through the anarchist mileu in the late-00s/early-10s.
I also think that the reason nobody talks about these types of ideas (building a culture, etc) is that most big-time anarchist thinkers who everyone respects already live where there's a bunch of anarchists (the Bay, PNW, New York, Montreal, etc). They aren't alone in their ideas, and so they turn inward and, in my opinion, grow stagnant.
If we want to spread anarchy, then I think these types of projects are worthwhile. And creating an anarchist culture is worthwhile too. It just has to dance around the trap of prefiguration.