To provide some context around this, one influence which could plausibly be behind a lot of peoples' criticisms of the BBC is the strong political resistance to the TV Licence Fee, which makes up about 3/4 of the BBC's income. The BBC's somewhat negative reporting of Prime Minister Boris Johnson (Conservatives) made the BBC quite a few enemies in Parliament, and there were subsequently no less than four different private members' bills from Conservative MPs and Peers to either greatly restrict the revenue that the BBC gets from the TV Licence Fee, or abolish it the fee entirely. Additional frustration comes from non-partisan criticisms of the heavy-handed behaviour of TV Licensing enforcement, which often intimidates people into paying unnecessary fees with misleading letters threatening criminal prosecution (my household has received such letters in the past).
How does all this affect the BBC? Well, everyone feels differently about the situation, but it's probably reasonable to say that the BBC has had a somewhat more pro-government bias since efforts to abolish the TV Licence Fee have begun in earnest, and the BBC tends to avoid some topics more than in the past. Its broadcasting has moved slightly to more profitable ventures, by developing programmes that it can sell abroad (nature documentaries with the slick new 'BBC Earth' moniker, Doctor Who etc.) rather than focusing on domestic educational content and reporting, which doesn't have a large international market and is therefore more reliant on TV Licensing income. It is likely to frustrate the Hacker News clientele with its vague coverage of 'Science and Technology' topics, which generally boils down to simplistic pop-science about black holes and the like.
It is true that some people feel the BBC is politically biased. but buts not just that.
The biggest problem is that people watch the BBC a LOT less than they did when there were a limited number of channels. Therefore people feel they get less benefit from the license fee.
Up to about the 1980s there were three TV channels, two BBC. The we got more and more new channels, and so the BBC channels became less and less important to people.
If the BBC were to disappear I would miss comedy and Radio 3. I have to TV so that does not matter to me - it does not stop Crapita sending me the threatening letters, of course. I think it is idiotic to outsource to them, and to fail to control their behaviour.
"it's probably reasonable to say that the BBC has had a somewhat more pro-government bias"
or less of an anti-Conservative bias? I think this is something people will argue about, and that is part of the problem.
The BBC has also gone downhill IMO.