The issue is (to me at least) not with the BBC, but with the expectations of the viewers. The both sides of the political spectrum don't seem to want unbiased opinion, they want opinions that supports their own views. Because of this, when they are presented with press neutrality, they instead see bias against them.
The only way they will be satisfied is if we end up with our own versions of Fox News and CNN, both so biased towards the right and left respectively, that it's difficult to take either seriously. The written press has been that way for years years, now it seems that the visual press is going to follow suit.