You don't "pay the Government". You pay TV Licensing, an non-government agency which collects the money and accounts for it before it goes to the BBC, which is not a government agency either. The government, through Parliament, lays down the legislative framework and sets the fee.
I am not sure why you seem to think it is "undemocratic". The government is elected by universal suffrage. If enough people thought the fee should be abolished, then a party would espouse that as a policy, and if elected would have a mandate to do that. They'd have to have an acceptable alternative though.