(Nor's the BBC for that matter run by the Government - it merely has requirements to be for the public benefit, to represent the UK, and a variety of other things, by law, and in return it gets money from the TV license.)
The OP may have been thinking of the Welsh 'equivalent' of Channel 4, S4C, which is funded from the license fee and advertising.
The term government is preferable because the term 'state' is vague to people outside the UK. Being a government run service does not mean the BBC is required to follow the ideology on the current party in power.
You're right about channel four though.
Actually... it's consumption of live TV broadcasts specifically (you can consume news, radio, and BBC iPlayer catchup without paying a penny), and it covers all live TV - even if you only watch, say, Channel Five online, and never access any BBC content, you have to pay the fee.
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.