Politics is about rhetoric, persuasion, spin and - if you can get away with it - outright lies, smears, and distortions.
Claiming that reporting is biased, when in fact it's merely objective but unfavourable, is an absolutely standard weapon in the arsenal of spin.
The only way to attempt to enforce a lack of bias would be to have independent editorial oversight - much like the journalistic equivalent of an independent judiciary.
The Board of Trustees used to attempt that, but the financial reality is that the BBC gets most of its funding from the government, and so is always vulnerable to financial pressures.
IMO lack of access to "clean" unspun fact-based reporting is probably the number one political problem of the moment. Knowingly lying to the public or attempting to influence opinion for personal/political gain should be as much of a crime as dumping asbestos in a school yard.
But currently the political system relies on dishonesty to operate. It's essentially built on denying facts and replacing them with expedient lies, emotional manipulations, and fake "explanations" of why things are the way they are.
So you can't solve one problem without solving the other. And that's a huge, huge challenge.
I think the simultaneous accusations of both left and right bias are because separate parts of the Corporation are biased in this way. If you sum those biases then yes maybe it sums to zero, but I'd rather the individual parts were not biased at all.