You don't seem to understand how the news media actually works. The New York Times is paid for by advertisements and subscriptions. The New York Times doesn't take money from special interest groups. There is no K Street lobby for Newspapers. You seem very uninformed.
>"The necessary role, and indeed the actual freedom, of the "free press" (in the traditional sense) in this democracy is in my opinion much overstated."
You don't seem to actually understand what the word "free" means in the context of "free press." "Freedom of the Press" is established in the First Amendment of the United States Constitution.
It allows the press freedom to publish any news they see fit without interference from the State. It allows the the press to publish news without fear of retribution or censorship.
The whole point in "Freedom of the press" is that it allows citizens to be informed in the case their government ever decided to grant themselves power that the people had not consented to grant them.
If you actually believe this is overstated or don't believe having that ability is essential to a functional democracy then you probably should spend some time reading a book or two about Civics. You also might want to look at the media. Or you could look at the current state of News media and democracy in places like Turkey, Russia or the Philippines.