Overall, over the decades, I find the Beeb make a reasonable fist of being unbiased. With the occasional ludicrous attempt to balance by allowing a fringe a voice. Some instances of which have been apologised for.
When an opposition gets into power they still complain, no matter who Paxman or Humphries has had a pop at lately. I think any preferences of the types of stories they run are a reflection of the times, and perhaps the sort of people who choose to go into the media rather than any overt bias. Thus for some they are viewed as biased centrist or centre left in always seeking balance.
More recently the BBC allowed Arron Banks to defend himself publicly against legal charges of election interference in a way which might well bias a trial.
The BBC might not be "government funded state propaganda" but BBC governors are chosen by the government.
The Arron Banks interview was indefensible, and I haven't yet heard a reasonable response there.
Since regulation was passed to Ofcom and the BBC board was created the government get to recommend only some. Not sure of exact number, but think it's a minority. Supposedly to further distance them from claims of state control.
I find the BBC really refreshing because they don't hyper sensationalize everything. The news is presented as if it's factual objective (not to say that it is, but it appears to be) and understated, much like the stereotypical English person.
When I watch British TV, I get a very definite feeling that I'm being persuaded to draw my own conclusions, whereas American TV, I'm blatantly being fed an opinion and told what I'm supposed to be thinking which I still after 18 years find incredibly irritating.
For all its faults, I still prefer the American system better. It's not better in every way... commercials (especially political ones) are a plague... but I would rather err on the side of Free Speech.
The British approach to TV is "This is what's going on. Do with that as you will." It's presented as if it are objective facts, calmly and with a distinct lack of emotion. The only time you ever seem to be force fed emotional opinions are by sports commentators and Band Aid concerts.