How is that different from being state-funded? Everything state-funded is paid for by the general public, through taxes. That's part of what being a state is: an organization that forces people to pay taxes and directs them to various programs.
Are you claiming that the TV license fee isn't a tax? It's money that the state makes you pay so that it can fund something.
Furthermore the state isn't in charge of administering it anyway, it's a civil matter brought about by the BBC (or rather the company which is subcontracted to enforce licencing). The BBC has the authority to do this based on the Royal Charter that governs it, that doesn't make it "state funded" or a "state broadcaster".
The enforcers work for neither the BBC nor the government but are subcontracted out.
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