Communicating empathy is certainly a skill. The assessor explained what she meant. It was about explaining procedures, distracting the child with a toy etc - all good to learn. Being a bit awkward and hesitant is not evidence that you are a psychopath.
I think you are wrong about the complaints. Chester handed over records of such things at Thirlwall and none about Letby. First is just tabloid gossip as far as I remember but would need to see it to say more.
Second was never identified as actually being Letby, and no, there was no complaint about this error. It was an interesting one, because the nurse in question, thinking the baby was dead, wanted to put the child in a cold cot (which preserves the body). If you'd just murdered a child, you'd be delighted to delay that as long as possible, since delays make the postmortem less effective.
Why would you complain about a nurse sharing a fond memory of your child enjoying a bath? Isn't that exactly how we are encouraged to offer comfort - sharing the good memories?
I'm sure anyone could be described as excited by an unsympathetic observer, when their job is to communicate as positively as possible with bereaved parents.
You only need a bit of empathy yourself to see that this isn't a catalogue of failings!