I don't know the US constitution, this is why I asked. My assumption is that running for president is not compatible with certain things, e.g. owning and/or editing media. E.g. could somebody in theory buy most of the newspapers, TV and radio channels, get on their boards and run for president?
Usually is the other way round , powerful people buy and manipulate the media so the one they want gets elected, and that way the can move everything behind the curtain.