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.