I don't think it's less neutral - your (non-standard) definition conveys the idea of government mouthpiece which supports your original comment but doesn't reflect the BBC's history or mission.
Didn't intend to convey it was a mouth piece for the government. But it's definitely official, and definitely a state institution, even the tail wags the dog a lot more than say XinHua.
"Public Service" is plainly a positive term though. It lends an air of altruism. Not particularly interested in what nomenclature is standard (according to who? 7
etc etc).