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Yes, and tragi-comically, the right wing nut jobs who claim that the BBC is full of communists don't realise that the BBC is actually helping them.
Or maybe a few of them do realise, but also realise that continuing to promote the "full of lefties" lie is still useful.
So in the end, nearly all parts of the political spectrum are dissatisfied with the BBC and the BBC itself is incapable of disengaging from the current political frame of reference.
(to be fair to beeb, same is true of other non overtly political news outlets too)
Amusingly this is a key argument to defend the BBC.