The innovation is that you aren’t born into a given regime and you can switch regimes.
This made the system capable of any number of regimes from the freelancer to coops to communes to partnerships to private corporations to public corporations to crown corporations to charities to not-for-profit trade associations to your kids lemonade stand to nonincorporated affiliated entities to manage the local block yard sale to Uber to open source projects to family trusts to whatever and so on.
However to manage a market function to extract petroleum cost effectively, it seems a corporation at the scale of BP, as a hierarchical corporation is the system so far shown to be most efficient? I presume anyway.