Most of what I’m going to say here is stuff I learned/thought about a decade ago. I went to the Porcupine Freedom Festival (Porcfest) in college, the largest libertarian get together in the US (it took over a mid-size campsite for a week). What I learned was hardcore libertarians tend to be anarchists. A lot of what I heard from others was centered around individuals being engaged in their own governance.

In one case that was a blockchain voting system where a person essentially owned their voting registration instead of getting registered with the state. The more interesting one was a presentation that put forth that the best organized groups are in sizes of eight. Therefore the best government system would be a council system in which individuals sort themselves into groups of eight, then choose one of them to be part of a group of eight groups, and so on until a country has built up a national council that directly traces each member to every individual they represent.

Based on those I think what most anarchists take issue with in our current world (of the US) is that the government comes out every four years to say “hey, here are some people to vote for, take your pick.” It’s a lot more top down than it might look on the surface, and anarchists demand a bottom up approach.

If you want an interesting portrayal of hard libertarian/anarchist ideals, I recommend The Moon is a Harsh Mistress by Robert Heinlein, which is about a libertarian revolution on the moon.