In this situation, heads of state and corporate executives are as much 'Average Joes' as kings and queens were a few centuries ago.

Yes, but a government and a corporation is much more than their head of state and corporate executive. If the actions of the government have a significant negative effect on a majority of the population, you can be quite sure that the government will be overturned at some point.

Who's going to do the overturning, though? If all you need is mental labor and capital, and AIs can supply mental labor for anyone... Well, the current holders of capital are going to have quite the advantage.