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Exactly.
She is trying to maintain her power and the party's.
BJ is such an amazing choice for so many political reasons. He appeases the Brexshit wing, while being one of the sacrificial offerings in the event of a Brexit failure.
Importantly giving him a prominent role that he is totally unsuited to gives him the opportunity to showcase his ineptitude to prevent him taking over the party. My guess is that she reasons that eventually he will go too far - a bit like tabloids and phone hacking;
celebs = "oh of course it happens, it's fine they're famous."
parents of missing children = pitchforks -
I suspect that Fallon was willing to go. It seems like he has a bit of a history so he was aware of all of the rest of the stories that might emerge. Sticking around would only damage his reputation further, and risk further eroding his standing in his constituency. And he has at least been able to cling on to that job.
I doubt it's that. It's just that May is powerless and rudderless. And everyone knows it.