That’s what I thought as well. I’d also note that their title is the “British Board of Film Classification” which is a lot closer to their actual role. Most films are passed uncut, and they really do act as an information service, not a censorship board.
The real target of the protest is probably the funding mechanism, which does appear to hit small, indie producers. But it’s difficult to see how to fund this otherwise, because if it’s paid for by general taxation, that would make them worryingly susceptible to government pressure.
Finally, interestingly enough, the power to ban or allow the showing of a film doesn’t even lie with the BBFC. Local councils have the final say within their areas.
