I don't think the wider media are specifically anti-Everton, we're not important enough to hate in their eyes. They're just pro-Liverpool, pro-Man Utd, pro-Chelsea etc. Pro-Top 6 but even then there's a pecking order within that.
Everton to them are part of "The Premier League" so our role in their mind is to provide games for the darlings to play in each season. If we get any individual coverage then it's usually uninformed broad strokes by whoppers who've done 15 minutes reading on us the night before at best.
You only have to look at how they all covered Sam Allardyce's time at the club. They haven't got the first clue about Everton or Evertonians.
I'll never forget as well on Sky's The Debate last year, Tim Sherwood sat there and said with an attitude that Everton don't play young players. Then when the host corrected him and started to say that we give more minutes to teenagers than any other top flight club he just interrupted him petulantly and said "not that I've seen they don't."
An actual pundit, in place as an authority on football, getting something as straightforward as that wrong. Even the most ardent Kopite wouldn't have said we don't play young players. lol