I'll admit this was a provocative and hyperbolic post, I was interrupted before I could polish my though. I'm British and I can recall plenty of times the BBC has bent over to accommodate the ruling party. The only one that springs to mind now is after the invasion of Iraq BBC radio stations refused to play Edwin Starrs 'war (what is it good for)'.
As I gather the matter, the BBC is subject to prior restraint under law, any time the Home Secretary chooses to invoke the relevant provision, as has been done for explicitly political reasons in the recent past with the Sinn Fein ban. That's close enough to "state broadcaster" for an ignorant colonial like me.
The ban on broadcasting the voices of Sinn Fein representatives (but not the content of what they said) applied to all UK media, not just the BBC. I'm in no way arguing that it was a sound decision, but it says nothing about the BBC being state run or otherwise.