Fox News is just an establishment safety valve. The fact of the matter is that "conservatives" in the US (as in the UK) have been conceding ground to "liberals" for decades, and the likes of Fox News (by encouraging so-called patriots to blindly support the useless Republican Party and by letting conservatives feel superior despite the fact that liberals have been beating them for decades) exist to make sure that nothing changes in that regard, and also (for good measure) to channel "conservative" anger in a pro-Zionist direction.
We have a similar sort of thing in the UK where the so-called "right-wing" press have been telling their readers to vote "Conservative" for years, ignoring the fact that that party has spent all that time moving in a more socially liberal direction.
Classifying media outlets as right-wing, left-wing or centrist is a convenient shorthand, but obviously leaves out some important details.
Really, those who consider themselves "progressives" should be thankful for Fox News for the service they provide.