How do anarchists propose handling public works, healthcare, etc?

Well, how we do it now?

During the Spanish Civil War, anarchists socialized many industries. What they found out was that you could just remove people from most management positions and continue work as before. Very rarely a manager was actually needed, and when it was they would simply elect one of themselves to fulfill the role for a while.

Talking about healthcare specifically, in many countries that have public healthcare, the system is already decentralized. Because it needs to be, otherwise they can’t properly answer the demands from their communities. Again, you just need to remove pointless middle-men and other workplace hierarchies (like physicians being more important than nurses), and stuff tends to get better.

To my (self-labelled anarchist) view, anarchism is not a complete political doctrine and does not propose an off-the-shelf solution for all problems. It is a general direction: reduction of coercion in society. Hopefully to the point where none is required. “But what about criminals, prison, armed police?” yes, we don’t have non-coercive solution for everything. We just know that where no-coercive solutions exist, we should favor these. And that’s already a pretty radical program.

public works

I am not sure what you specifically project in anarchism, and whether you are talking about the actual construction work or the decision-making but these typically tend to work better when all stakeholders are involved in the project and there are already public works cooperatives out there.

healthcare

You mean health insurance or actual medical care? In my country (France) actual medical care is already done by a staff of people who accept a level of stress and a low pay they could easily escape given the qualifications required to work in a public hospital, but cling to it due to their will to accomplish a useful work. Remove administrative hassle and the need to pay for a right to live and they will happily work for free there:

Same applies for most teachers, researchers, caretakers, farmers, that I know.

There’s an entire essay just for how would anarchists do public service like healthcare. tl;dr, healthcare wasn’t even provided by states in the early modern period, it was provided by mutual aid societies. These societies had to be divested of power for states to monopolize the provision of services.

Community banking and funds mostly. Health-care similarly. It’s actually not really such a big problem to organize these things, so I always wonder why people give this as examples for why Anarchism wouldn’t work. If anything, large bureaucratic state systems make these more difficult and expensive.