I recommend https://theanarchistlibrary.org/library/david-graeber-possibilities

Violence is in fact unique among forms of human action in that it holds out the possibility of affecting the actions of others about whom one understands nothing. Any other way one might wish to affect another’s actions, one must at least have some idea who they think they are, what they want, what they think is going on. Interpretation is required, and that requires a certain degree of imaginative identification. Hit someone over the head hard enough, all this becomes irrelevant. Obviously, two parties locked in an equal contest of violence would usually do well to get inside each other’s heads, but when access to violence becomes extremely unequal, the need vanishes. This is typically the case in situations of structural violence: of systemic inequality that is ultimately backed up by the threat of force. Structural violence always seems to create extremely lopsided structures of imagination.

As I understand anarchism, the idea is a society where human culture becomes powerful enough to overcome and replace this sort of violently imposed top-down structure.

My current understanding is, destruction of current system of government (violently or otherwise) followed by abolition of all law. Following this, small communities of like minded individuals form and cooperate to solve food, safety, water and shelter concerns.

I think your main mistake is to get this backwards; the mere destruction of government and law doesn’t by itself effect the formation of anarchism. You need a culture with enough utility and resilience to replace it and endure without falling back on the crutch of structural violence.

The book I linked goes into some detail considering what that might take, focusing on the example of the nearly-anarchist society of 1990 Madagascar, where technically they were under the rule of a formal government, but in practice almost all governance was independent from it and driven by their unique culture. To summarize a little from memory, ambitious people basically aspired to be liches, with living supporters conducting regular rituals involving their tombs and bodies to avoid getting cursed, because having a prominent place in a reputable tomb after death was the only path to be considered an important person. But the main way to get such a position was to provide for people enough that they would become able and socially obligated to maintain your place in the tomb. There’s clear social utility there; achievement materially depends on positive contribution.

If it is the case that the concepts and relationships that define society and how we behave are essentially feats of imagination, then it should be possible for this force of imagination to itself be the basis for holding things together, rather than forcing it into artificial molds defined by violent hierarchies. What’s needed for that to happen is to sufficiently develop cultural imagination as a technology that it can build systems that stand up to the pressures they need to bear, that currently get handled through destructive shortcuts that treat people as things.