We do have a lot of public works and utilities run by private companies; in some contexts it works mostly fine, but government oversight and regulation are crucial to avoid predatory monopolies (see railroads 100 years ago, lots of electric companies, cable). Other times the experiment has gone horribly, like for-profit jails (google about the scandals where it was revealed that some judges were getting kickbacks for sentencing juveniles to detention facilities).
w/r/t the OP's original scenario, where companies have a share of government control based on market value (I get that policy follows money now, but not nearly this directly or precisely), I expect it would be unstable turmoil. Think of all the hundreds of thousands of people employed and billions of dollars that scrupulously watch the Dow and every micro-percentage of movement, which can shift wildly within hours. Would it be like that, but with shifts in government power, and all the law enforcement and brute military strength and nuclear codes and foreign policy power that goes with it? Could a billionaire "angel investor" prop up the value of a particular company and run a puppet government? What does 27% of federal power look like? Votes in congress? Who controls the executive and the judiciary?