That is to say, the BBC is supposed to be run for the benefit of the United Kingdom and its citizens, not for the benefit of whoever happens to be the elected leader at the moment.

TL;DR "run by the state" = run by the executive appointed by the trust appointed by the monarch.

The government doesn't have direct operational control. The BBC executive is appointed by the BBC Trust. Trustees are appointed by the monarch in accordance with the recommendations of the government. So, beyond the selection of the trustees, the BBC is independent of government.

In the past the BBC has been somewhat anti-establishment especially during the Thatcher years. Recently its been a bit more pro-establishment especially as the BBC's charter is to be renewed in 2016 and during which issue of funding will come up.