The BBC hasn't been cut in the name of austerity. It isn't funded by general taxation it's funded by the licence fee, which has only risen, however not in line with inflation. This is basically because nobody wants to pay it, and indeed a huge number of people have stopped paying it.
You're mostly correct - I was absolutely generalising, using internationally known examples (like the BBC), to be more comprehensible to the international audience here about cuts to public arts spending.
The BBC was a silly example of me to use (as was 'public radio'!) because you're right - it's mostly not funded by general taxation.
That said, the license fee is directly tied to inflation (via the CPI). Whether you see that as a correct measurement is another question. [1]
The BBC has also historically been funded partially by taxation. For example, taxation provided free television licenses to over-75s, until 2020, when that funding was cut, and the cost of doing this was passed to the BBC itself.[2]
But yes - quibbles aside, a sloppy example for me to have used there. Your sibling comment has some great sources which better illustrate changes to arts funding, compared to the off-handed ones I initially provided.
[1] https://www.tvlicensing.co.uk/faqs/FAQ23
[2] https://www.ageuk.org.uk/northumberland/about-us/volunteer-f...