Because now you're proposing that people be charged for something even if they don't use it and don't want it.
Not everywhere it's all about the individual and their whims. If the majority (well, through elected representatives) wishes to have and sustain a public good (whether a public broadcaster, or national health service, or arts grants, or whatever), individuals have no other say on the matter.
Some countries believe in public goods, and in maintaining through taxes infrastructure to make citizens more educated, informed, cultivated, etc. If someone is a brut and could not care less, they don't get to dictate to the nation not to have those services.
E.g. the German example of the tax for the national broadcaster a fellow commenter mentions:
"Any household in Germany is legally obliged to pay this quarterly fee, regardless of whether or not you watch the TV channels or listen to the radio stations covered by it. It also covers media consumption online via on-demand services such as media players, streaming services accessed online via computer or smartphone, as well as in-car audio. Shared households are only required to pay this per household, so 4 students living together for example would only be liable for paying the fee once."
In Australia we fund the ABC and SBS through tax, which have charters requiring a certain amount of locally produced programming, and serve as publicly owned news media (which I think is in the public interest so long as the government isn't exerting too much in the way of influence), as well as programming that could be considered too niche to be profitable otherwise.
When I was a teenager SBS was how I was exposed to some really good stuff I wouldn't have been able to see otherwise because it was anime or foreign language films.
Edit: It took a reread for me to realize that you were actually arguing in favor of it. I do see your point about positive externalities, but I'm still not convinced it's a good trade off overall.
Well, life is more than the base "requirements". Humans are not wild animals that only understand bare utility, so some societies/states can acknowledge that and also provide some "non-requirements".
Besides TV (and cultural products in general) is not universally viewed as mere "entertainment", but also as information, culture building, citizen development, and so on. Some nation states do trust themselves to have capable people to provide quality programming -- whether many of their folks just care to watch reality shows and entertainment or not.