You've got to remember these sports shows are just mindless chatter for the most part. The idea isn't so much to be "right" as to be saying something, anything, just to fill time. And if you can set two people with opposite opinions against each other, so much the better, then you get the debate angle. It's just entertainment, kind of like The View or Oprah for men.
As for Skip, as I said before, he's such a cartoon character he doesn't bother me. I have no doubt he dislikes Rodgers, but there are times he can't keep the smile from creeping onto his face when he's bashing him. He knows he's full of ****, well sometimes he knows.
I seem to recall someone saying that the reason Skip hates Rodgers is that he was a big Favre supporter, and went on at the time that it was a big mistake for the Packers to move on from Favre to Rodgers. Rodgers kind of proved him wrong, and he dug in from there. I don't know if that is true or not, because I didn't watch First Take back then.