One of the important things to note is that the US absolutely dominates world media, from TV to movies to music to everything else. For countries that are linguistically compatible (UK, Canada, Australia, etc) this is extra-true. You get into a weird situation where the bulk of the entertainment being consumed by the populace is foreign-made, and there's increasingly less room for expressions of local culture.
The funding of the CBC in Canada for example is less about making you watch the CBC but more about state funding to create local arts and culture in an environment where foreign culture dominates. It's about making sure "Canadian TV" and "Canadian music" even exists (sorry about the Bieber).
I can't speak for the BBC, but it's also important to note that unlike public broadcasters of the US like PBS, the CBC is not a giant money pit. It creates programs that are highly successful, generate revenue, and have viewership competitive with major American programs. The CBC is a mainstream cultural institution in a way that the PBS can only hope to be, so it's not as if taxpayer money is being spent to create shows nobody watches.
The entertainment isn't produced by the state though. The BBC is owned and funded by the British public, not the state. The government isn't directly involved in the BBC, although "BBC Trustees are appointed by the British monarch on advice of government ministers"[0]. Other European countries have similar models.
Besides, and I understand that this is a contentious issue, I think that there are certain services that all members of a society should pay for, even if not everybody uses them. My taxes pay for roads even though I don't own a car. Bear in mind that the BBC also produces news and a lot of other content, all without ads and independent of commercial interests.
[0] https://en.wikipedia.org/wiki/BBC#Governance_and_Corporate_S...
Yes, it is messed up.