The state-controlled CPB exerts significant influence via funding and the FCC via regulatory reach. This isn't theoretical. Whether that's "direct" control or not is in the eye of the beholder, but people have accused media in other countries of being not independent under similar circumstances.
Personally I think anyone can live in their own fiction as long as they are not hurting anyone else, but to consider a strategic resource like the media as being truly independent in any country is naive. You're literally missing an entire world if you do not look outside your home country.
> They aren’t directly controlled by the government though.
No. They're controlled by multinational corporations with clear profit motives and conflicts of interests and thus tend to be biased in favor of those industries.
In China you have state dominance so you have state controlled media, in the U.S. you have corporate media. That's not independent media, just the other side of the same coin.
And if your argument is "they don't control the army", the military industrial complex means they don't have to do so directly.