The BBC should aim to represent fully the nation who fund it, including those with views others would find abhorrent, if those views can be expressed in accordance with the law.
The BBCs core purposes as I have always understood them are Inform, Educate and Entertain.
If people cannot express their views in such a way as the law allows, they shouldn't be able to. God, this really isn't complicated.
You are again confusing a news department with a polling consortium.
You can't 'represent' the views of selected individuals including many who don't know their arse from their elbow and simultaneously inform and educate, though you can certainly provide a ghastly form of entertainment.
When a news organisation trips merrily across the line from informing and educating into garnering ratings by means of providing entertainment, when serious debate is ditched in preference for circus-like spectacle, it renders itself irrelevant as an informer and educator.
If you can't see the difference between provision of free airtime to clowns and the provision of serious, factual news and impartial analysis then no wonder the UK faces disaster and dismemberment.