Sorry but from your comment I get the idea that you have the popular misconception that anarchy just means “no government”.
Anarchy is the destruction of all heirarchy including the state, this includes class heirarchy. Anarchy is very similar to utopian marxism, just skipping the “dictatorship of the proletariat” part.
Anarchy is the rejection of unjust hierarchies, not all hierarchies. Certainly a parent would have authority over a small child insofar as that’s reasonably justified. Similarly, some expert may be elected to a position where they’re democratically authorized to make specific decisions so we don’t have to vote a thousand times a day about specialized matters.
On the whole, anarchism just means a lot more democracy.
If it’s real anarchy, there can’t be billionaires. One of the central tenets that almost all anarchists agree upon is that capitalism and the state support one another, and so both need to be demolished simultaneously. Destroying one while preserving the other will, as you point out, lead right back to the old system. We see the similar but inverse situation in Russia and China, where attempts to destroy capitalism with a strong state also lead back to oligarchy.
You might be thinking of ancapism, which is widely rejected by most anarchists and not considered to be a real part of the movement.
Their point is that you'll find it difficult to "demolish both" the state and concentrations of wealth and power "simultaneously" when you're fighting through billionaire-owned militias. Good luck.
So in an anarchist society, how do people settle disputes? There can't be a law without some form of governing body to enforce the law. Seems like a might makes right would bubble right back up to the top.
And the very far left anarchists, the soulists, reject reality itself, and don't believe in a real anything. But soulists also hate ancaps and recommend giving them all a radical headectomy.
Anarchism can work (it is working everyday currently). It just does not look the same everywhere. The idea would be for many communities to build strong mutual aid such that they don't need to rely on an oligarch. We need to move away from capitalism and form communities that can associate with each other and help one another. It isn't so much of a one time revolution, but a process that needs to be worked towards over time.
"We all", so it's not an anarchy?
I mean the second you get everyone agreeing on something and acting it out, it's no longer anarchy.
It's a commune of VERY like minded people that value keeping to them selves, but militarize the second someone threatens their way of life.
And unless it's a hive mind, such a force will naturally find some sort of organizing body or fizzle out.
Switzerland is a lot closer to what you're describing than anarchy.
The state is basically already a billionaire's union that allows them to get a group discount on security, in order to protect them from the rest of us. They even make us chip in for it too.
Billionaires already own the police, which number over 700,000 in the US alone and the national police budget would be equivalent to the third most expensive army in the world. If this is already the state of things, how could we blame that on the anarchists? This argument effectively boils down to "we shouldn't have a revolution because the rich would have a monopoly on violence", which we already know to be the case in our current society. So in this way, nothing at all is holding us back. The worst case scenario would be a preservation of the status quo. Even if the Proud Boys, III%ers, Patriot Front, Oath Keepers, etc. all combined into a giant mercenary group, it wouldn't be even close to how the cops already are. They're one of the most militarized police forces in the world with access to heavy equipment such as assault rifles, chemical weapons, acoustic weaponry, MRAPs, and much more.
If the revolution ever comes, we'll just have to take all the billionaires to the Ipatiev House. A revolution would already have no recourse but to defeat any police opposition anyway, so there really is no difference whether the billionaires are around or not. Those billionaires, if they have any sense of self-preservation, would be smarter to take their money and simply flee abroad.