If we're talking about how to allocate resources, then this: "The way to show government should intervene into personal lives" is a super loaded prompt. You're assuming that society's resources are personally claimed, which is even more radical than assuming they're privately claimed.
It's difficult to think of a system where the state doesn't intervene in how resources are allocated. In our current system, the state facilitates and protects private ownership. Property is 90% of the law. If you're looking for a place to stay and you come across a foreclosed home, if you squat there, the state will come and kick you out on behalf of the bank that owns the house. When a group of Indonesians make a flat screen television, it doesn't go to the workers or the factory itself or even stay in the country; the product and its bottom-line profits are allocated to a foreign entity. This is a great deal of what the state oversees.
In anarcho-capitalism, the state goes away in only the most pedantic sense. The military and police are replaced by mercenaries and mall cops, but I guess at least the facade goes away that they're not acting on the direct behalf of private interests?
In vanguard socialism, the state itself owns the resources and allocates them directly.
In social democracy, the resources are primarily allocated via private ownership claims, but the state taxes enough that the government itself has enough resources that they can come to democratic decisions on how to allocate that segment of the economy.
Even in most forms of fairytale anarcho-communist societies, resource allocation is ultimately determined by some institution. In anarcho-syndicalism, for example, it's basically the unions: syndicated groups of laborers jointly determine how they're going to use the means of production and to what use they'll put their labor.
So it's just a matter of how you define the terms in your OP. If you think it's self-evident that someone having a picnic in a Baron's garden is an infringement on the Baron's personal liberties, and so having the police show up to kick them off isn't a government intervention, then you're going to demand an explanation if the Baron's estate ends up being used as a public space. If you see the police dragging a man away for having a picnic on an unused plot of land as a government intervention, then you'll see it exactly the other way. So having the conversation on those terms isn't terribly useful.
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