I haven't heard of the false equivalence before, but it doesn't sound like this. There is that popular fallacy though, but I'm not aware if it has a name, where people think that the truth is always in the middle (the golden middle).

And you could say that this is it, but given the strong polarization and extremism around topics that involve politics it might be a good measure in this case. (Of course, depending on what exactly 'everybody' means.) By definition extremists of either side will see independent actors as hostile. But, of course, this is just a measure, not a proof.