I don't necessarily think these things are related nor that they've been accurately reported.
It seems with the David Attenborough thing there was a series commissioned and then an additional 'episode'/ film was produced with a number of charities which was not part of the series but intended to be released alongside via iplayer. And there has been no change to that plan. Given most people don't watch live any more anyway, I'm not sure really what difference it makes. If they were trying to 'silence' debate surely the easiest thing would have been not to platform this film at all?
Fiona Bruce - her comment was badly worded but it was live, not scripted and I think she was trying to clarify what had happened rather than pass a value judgement. It wasn't good but It seems a bit harsh to 'kill the witch' over a one off, live remark. Has she clarified what she said at all? I highly doubt she would have intended to minimise male violence. I also think, given it was a live show, that it was more likely to be her getting it wrong rather than a command from on high from the Beeb management to minimise male violence.
With refugees - there is no nuance in the debate and local people are being ignored. If GL really wants to support the refugees how about he builds a centre for them where they can be properly housed while being processed near his home? He has enough money to do so several times over. But no, he'd rather say not do (unlike other footballers who actually DO things with their wealth). I don't think the government are getting it right but I think the extreme polarisation of the debate so that schoolgirls who say male refugees have sexually assaulted them are cast as right wing bigots doesn't really help anyone (including the refugees) much.
Having said all this, I gave up on the BBC ages ago (their choice not to safeguard children from Savile - despite very many people speaking up- sealed the deal for me once I became informed about it) and haven't paid my licence for years - you just sign a declaration and it just means you can't watch live tv (which I didn't anyway) or iplayer. Anything I want to see on iplayer is usually available on another paid platform eventually.
If you don't like the BBC, just don't pay for it or watch it anymore. Many other methods of obtaining news and entertainment are available. Alongside other sources, I quite like Al Jazeera for news - provides a different perspective which I think can be very beneficial.