[interestednovice]

The BBC is not a state broadcaster, no, but you could still fund it via another tax if appropriate safeguards were in place. A separate hypothecated element of Council Tax, that went straight to the BBC, for example would do this.

There is no reason the government couldn’t “collect” the tax on behalf of the BBC. This wouldn’t politicise the BBC itself as such an arrangement would be agreed far into the future, so would continue regardless of who was in government.