The fact that we can't agree what the BBC's bias is, would suggest that any bias it has is fairly trivial. I mean, hands up anyone who doesn't agree that the Telegraph (say) is Right-biased and pro-Brexit? And yes, the remit of the TV news means that any bias should be minimal, compared to newspapers.Most people complaining about political bias at the BBC generally mean: "how dare they question my bias". If the BBC have a bias, then it's the tendency to be managerialist, and generally conservative with a small c. Left or Right, not so much.
And yes, the party in power gets a slightly easier ride because (surprise) they control the BBC purse strings. The miracle is not that the government of the day gets away with a bit, but that doesn't get away with a lot more.