I had a look at the facebook petition page and came across this thread. It seems pretty sensible to me and I can't see any faults with it. In the case of the UKFC, I've heard little strong arguments in favour of it.
Thinking about it, how many good/great films has Britain produced in recent years? They're anomalies, basically. As the thread points out, many of the 'triumphs' are part American productions. You could point out films like Red Road or Fishtank as the council taking risks- but it's hardly likely someone who had just won an Oscar for her short film was going to be denied backing.
The idea of bringing back the BFI production board sounds interesting- apparently they focused on more experimental and innovative films, and brought through new film makers. Many of the councils films seem pretty mediocre affairs. Just looking at the trailer for 'Made in Dagenham', it appears they have diluted an interesting story into a middling, 'feel good' film. Not that this is terrible or the film will be terrible-just that it's a tired formula that Britain has become saddled with.
Film makers such as Ken Loach, Mike Leigh and Peter Greenaway have to get funding from other European countries to make their films. The BFI production board was supposed to be good at supporting these kinds of film makers.
Maybe I'm completely wrong. i don't know the ins and outs of the situation, I'm not an industry insider. Just the feeling that the UKFC was a middling organisation and we could do better. The thread I linked to makes a convincing case.