consultants just happen to single out a particular nurse...who just happens to be on duty during the deaths
These aren't independent things: they singled her out because she was on duty during four deaths.
and juat happens to have stacks of notes at home...
But you'd list anything unusual in her life/home here, so all you can say is, she just happens to have something unusual. And having something unusual in one's life/home is not that rare, even if having a specific unusual thing is. "Just happens" implies unlikelihood of finding what was found with the search that you did, but if you searched for "anything unusual", finding something is not actually unlikely.
just happens to have written weird things in a diary
Only after getting accused, so it didn't "just happen" independently of the accusations. Mining diaries for "weird things" is error-prone anyway. E.g. Lucia de berk 'wrote about a "strange compulsion" and a secret she would take with her to her grave. On 27 November 1999, the day an elderly woman died in her care, she wrote that she had "given in" to her compulsion.' Kathleen Folbigg wrote in her diary about the daughter she was accused of killing: "I knew I was short-tempered and cruel sometimes to her, and she left. With a bit of help."
and just happens to be a bit of a loner and reportedly a bit arrogant at times
She wasn't a loner, and "reportedly a bit arrogant at times" is quite common. And again, you'd list any arguably odd trait here, so it's not at all surprising to be able to find some. "Just happens" implies something is unlikely to be found by chance, and if you don't define in advance what you search for, finding something is very likely.
just happens to be selective in her testimony
How, exactly?
just happens to not seem to be overly vocal in protesting her innocence
She was quite vocal. E.g. her Head of Nursing said: : "In weekly meetings to check on Letby’s wellbeing after she was removed from the ward, Rees described how she “cried in my arms”.
“She was crying … very distressed every time we met her, saying: ‘Why is this happening to me?’,” she said. “She kept saying to me: ‘I am not going to let them drive me out of the job that I love. I worked hard. I’ve done nothing wrong.’ That’s what she kept repeating to me. It was harrowing.”
Rees added: “People say she’s not emotional. Trust me, she is emotional. I know that they say psychopaths are clever. But if she was acting she deserves an Oscar because she was so convincing.”
"