That article doesnt say anything about brand getting her fired
Brand and Chiles reported her to the higher ups and she was sacked for it, it isn't directly Brand getting her sacked I agree.
It's mudslinging where non is needed. The hypocrisy has been played out though. Jokes, nuance, context have all been used by the same people who not only ridiculed said terms but ramped up the weaponising of the police, their employment etc for what amounts to edgy jokes.
I don't think she should be investigated over it, it wasn't a great joke and considering it was aired on Radio 4 it really wasn't pushing any boundaries either, it was more along the lines of backpatting humour which is not even punching up. IMO comedy needs a new revolution like the 80's but thats another conversation.
The hypocrisy is that some of the vocal support for Jo did not offer the same support for the others making edgy jokes (apart from the comedians who have come out in support) because they are basing it along tribal political lines.
Its about holding the left to the same standards that they themselves have wanted to impose.
She has apologised, the BBC has removed the content but you can guarantee it will get brought up again when someone right leaning makes a joke.