One of the biggest misunderstandings about anarchist philosophy is the idea that it's about being "against the government". Right-libertarians and ancaps love cultivating this one because it allows them to present their overtly-fascist ideology as somehow "liberatory". Liberals love cultivating the same idea because it helps them maintain the myth that there is a bright line of separation between the government and the rest of capitalism, that there is some sort of mass power to be found in liberal democracy.
The thing to understand is that "the government" does not exist in a vacuum. When anarchists talk about the "capitalist state", we do not mean "the government". Capitalism is a totalizing ideology and the capitalist state is a complex made up of all of the organs and mechanisms of power wielded by the capitalist class against the working class. The government per se is just one part of that, a historically contingent formation for wielding violence against the working class with popular consent. Capitalist media is another part. The system of wage labor is another. The Westphalian international state system is another. And so on.
This also means that when we talk about dismantling the state, that does not mean just eliminating The Government and leaving everything else the same. Capitalist power doesn't derive from the government, and indeed getting rid of it would (as right-libertarians and ancaps are so eager to see) simply turn capitalists into warlords; it would be an inconvenience that they had to wage war directly against the working class instead of having an organ with an accepted monopoly on violence, but just that: An inconvenience. We must destroy capitalism to destroy the state as it exists.
This also means that it is not inconsistent to support public services as an anarchist. Public healthcare strengthens the working class. So does public education, so does a robust welfare state. These are all extremely important for mitigating the harms of capitalism. It should be obvious that if there are ways to bend the capitalist class's own tools against them, that is entirely supportable. We have to look at what is the most effective approach to build enough power to do it, and with an understanding of how the state complex is constructed this also means developing a critical understanding of what of these functions of the government can actually be bent to socialist ends.
