You're conflating the use of the word public here.

Public corporation: open to any old Joe or Jane that wants to become a shareholder.

Public interest: needs of a society.

Walmart, Google, etc. should not have to pay for lighthouses or sewer connections down a street, nor a legislator's fact-finding trip to Shanghai.

No, I'm not. Corporations are pure creatures of state law, and "public corporations" are partially creatures of federal law (shareholder rights are protected by federal law). The modern public corporation could not exist without the government's explicit blessing, and it is quite likely that huge corporations like Google could not exist without the benefits of limited liability. Without using corporate forms to compartmentalize and "firewall" liability, it is very difficult to scale organizations up in size.

Could a corporation exist without the implementation of state law? What is a corporation if not a group of people working within a hierarchy?

Companies could exist without explicit state law, but corporations could not. A company is a group of people working within a hierarchy. A corporation is that + limited liability + fiduciary duties. As a practical matter, lack of the latter things makes it difficult to scale up organizations where the people involved don't necessarily trust each other.

Remember, all of these complaints about corporate taxation would disappear if companies would simply organize as say partnerships, without the legal benefits of the corporate form. There would be substantial tax savings to this structure, because partnerships are simply pass-through vehicles (the money belongs to the shareholder immediately when it is earned). They don't do this because big partnerships are impossible to manage.