I concur with your opinion that it is very likely an agreement between (most) Democrats to spurn any invitations to appear on Fox News as a way to avoid being asked questions they can't or don't want to answer and thus, allowing them to make the false claim that Fox is 'biased' against Democrats because few Democrats appear on the network. Cute.
Fortunately, Fox News does have quite a few Democrat/liberal guests, usually former Democrat party or Clinton officials and some left-wing commentators, such as Dr. Mark Lamont Hill, who is occasionally borderline loony but who is refreshingly forthright about being a liberal. Most of the Fox News shows have liberal guests and the pure commentary shows, such as the O'Reilly Factor, have representatives from the liberal/Democrat side on almost every show.
The false accusation from the left that Fox News is 'biased' stems from the obvious fact that they present both sides and, yes, they 'lean right' a bit but nowhere close to the way the 3 TV networks, MSNBC and CNN 'lean' left. Fortunately, Americans are catching on to the 'mainstream' media's leftward bias and abandoning it in large numbers. Meanwhile, Fox News is growing by leaps and bounds. That indicates a shift in viewing habits and validates the premise that Americans are sick of one-sided 'reporting'. Fox News is careful not to go too far 'right' and so they disappoint some conservatives, like me, sometimes, but they retain their hard-won credibility by doing so.
You and Citizen M hit the nail on the head. Well presented.