I agree that no prisons should be privately run. They are organs of the state and should be run by officers of the state, not jumped-up, ill-trained security guards.
BLM is a wider organisation and message than BLMUK. In a sense it's not even an organisation at all, or at least there's no overarching structure beyond a desire for justice. Yes, I missed some of the aspects of one specific "chapter" of it.
There's always more research to do, of course. But I'm satisfied that the underlying causes of BLM are worth supporting even if their more extreme wings may not be.