Wait - I thought "BBC" was "public media", not "state-run media"? Or is that the joke...
It's the closest thing we get to "State-run media" (together with Channel 4, actually, and I suppose S4C is also part-funded, etc), but that's only because the state controls the purse-strings and have
strong opinions about how they get to spend money from said purse.
It doesn't stop the BBC criticising the current government (whichever colour), and just about anyone else it thinks it ought to - including the BBC itself in some fairly extreme cases[1]. People on the left say that the BBC is too right-wing, people on the right ay the BBC is too left-wing, people in the middle say the BBC doesn't represent the middle-ground.
Every government (but especially the Tories) squeezes the BBC through the power it has to control the funding (but then the Tories also try to do the same with the police, transport, basically whoever isn't toeing the line or making them look bad - and Labour has been known to be awkward in that regard, too, while the Lib Dems haven't had much chance, except for backing down on Tuition Fees promises!) and the current 'ultimatum' is that the next review of funding "Will be the last!", with apparently no practical plan for how the funding model can actually be changed.
Every other media/entertainment outlet envies at least some aspect of the BBC, too. Though they are not beholden (albeit mostly ill-disciplined) to the government, they have their own interested-parties doing more particular brands of influence (e.g. ITV is a martyr to its advertisers...).
[1] It is not unknown for the BBC News to announce that, in relation to some BBC-related news-happening, "We tried to get a comment from the BBC, but no-one was available...". This being the the bit of the BBC that is in trouble refusing to comment, rather than the bit of the BBC that is reporting the trouble refusing to try to get a comment...