Well, let's do a little switcheroo here:
Someone has to sit near the top of the tree, making decisions whose impact has to be accepted by the shareholders, employees and customers of a company, and communicated to the governments of the world. Not all products and services can be implemented by a small mom-and-pop shop. Some things need scale. And not only that.
Corporations are far (far!) from perfect, but at least they aspire in the right direction (where "right" is defined by the zeitgeist of those people they service above, and so by definition what behavior will a corporation tend towards).
Governments are beholden to no-one but their sponsors and shallow populist trends they use to rise to power by exploiting the large distance between voter and representative, often nothing in common to discuss, so the debate is kept low-level, basic, and full of the same lies repeated over and over again to slightly different audiences every 4 years or so. Politicians are demonstrably not aligned with what is best for the populace, but rather what is best for them clinging to power and servicing their behind-the-scene sponsors (or in some countries, public lobby). We can't change that property of governments easily, but we can, in theory, hold companies to a higher standard than "do whatever the highest bidder of our party says". We can vote with our money, and vote with our speech, and boycott a product or a service much more easily than we can outvote an incumbent party or a law, when they betray us. As we see in France, protests do nothing. Big once we have aligned corporations to us, they can represent our voice in front of large, cumbersome governments, and bring balance back to us. So, I'd like corporations to stay near the top of the tree, and not make this a government autocratic monopoly of control.
P.S.: BTW when you say "companies are beholden to nothing but the markets..." I hope you realize... we're the markets. Not any less than we're the voters.