The BBC provides a service for the public good
The BBC provides a unique service that can be consumed by everyone equally.
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There are some services in British society that are non-excludable. The can be consumed and enjoyed by everyone equally with no barrier to entry. They also serve a public good. These institutions must be publicly-funded, regardless of cost or burden to the taxpayer.
Another example of a non-excludable service that is provided largely for entertainment purposes is the Royal Family. In the UK, there is a sizeable segment of the population that does not agree with the concept of the monarchy. At this point, the monarchy is entirely symptomatic and their public function goes little beyond entertainment. But it would be inconceivable to make the financial support of the Royal Family based off of a subscription model or voluntary donations. They are a national, non-excludable institution. They provide a unique service that cannot be replicated or found elsewhere, and everyone in the country can enjoy the entertainment benefits they offer. The BBC functions is very much the same way and should be funded in the same way, through the public purse. 
The BBC was once a non-excludable service that served a unique function. It is not anymore. Once the BBC provided a service of essential public need. It was informing the population and providing a unique service of a public good. Now, with a host of other media outlets providing news coverage, the BBC’s public worth is severely diminished. Additionally, the Royal Family is public-funded, but not through a mandatory Royal license fee. It comes from our tax revenues. Nothing would stop the BBC from being funded in the same way, removing the arbitrary and inefficient license fee.
[P1] Non-excludable services that offer a public good should be publicly funded. [P2] The BBC is a non-excludable service that offers a public good. [P3] Therefore the BBC should be funded through a mandatory license fee.
Rejecting the premises
[Rejecting P2] The public worth of the BBC is limited in the modern media landscape. [Rejecting P3] There are other formats for publicly funding the BBC, including through general taxation (like the Royal Family).