I know that is the theory but you will get just as many criticisms from the left as from the right about the BBC.
And as "the" public broadcaster it is part of the public, ie reflects what are seen to be generally accepted views.
ie it doesn't sit back and say we are the ones who can act with true impartiality, because we are untouched by the public mood.
If you remember they go told off in the early days for giving a disproporionate time to Farage during the whole Brexit era.
But that was driven by public interest.
Obviously, I think they like so many institutions have been and still are totally Stonewalled. And even now it is hard to believe how many institutions were and still have this embedded in their practice.
In fact in the early days the BBC was very much not about impartiality but about making sure the British public were constantly reminded of what a small elite group thought British (but mainly English) values should be.