He agreed the mother's recounting of events would be "very valuable" for use in training staff.
The case, before Coroner Brigitte Windley at the Dunedin District Court, is expected to conclude today.
Millar reiterated the view given yesterday by leading paediatrician Professor Stuart Dalziel – that the two-year-old girl should never have left the hospital.
Mr Millar began by apologising to Hineihana's family.
"We acknowledge we should have recognised Hineihana was very unwell on the second visit to hospital and she should not have been sent home," he said.
"Failure to recognise the severity of Hineihana's illness led to a series of events that would have been deeply distressing to the whanau and would have increased their pain, suffering and grief that they undoubtedly experienced. We remain truly sorry for this."
A review by the Health & Disciplinary Commissioner (HDC) found the consultant who discharged the toddler without physically examining her had breached the code of conduct, as had a nurse who later spoke to Hineihana's mother and failed to give appropriate advice.
Another doctor who saw the girl was reprimanded for his actions and the "quality of his documentation".
Millar said there had also been an internal review, the recommendations from which overlapped with many from the commissioner.
He told the inquest of the various measures SDHB had implemented in the wake of the incident but said the circumstances which led to Hineihana's death were "unique and tragic".
If the coroner was going to make recommendations to avoid a repeat outcome, he suggested they focus on re-presentations at emergency departments.
"The critical point was the second visit . . . when the course of events may have been changed."
Millar said it was imperative doctors listened to parents.
"When parents bring their child back to ED they bring them back because they're concerned and it's really important in those circumstances . . . to go back and start from the beginning and not make assumptions or continuation of what was previously thought at the first attendance. After all, the parents know their child best," he said.
"We always have a risk of what we call 'cognitive bias' . . . people start to form an opinion before they have full information based on things they have heard or been told or read – it affects all of us. As clinicians, we've got to try and avoid being caught up in that."
The enquiry previously heard Hineihana had been discharged despite a junior doctor and nurses having worries for her wellbeing.
Millar said the health board had worked hard to eradicate a culture where staff felt like they could not challenge decisions.
"I can see in this there were people who felt concerned, who maybe didn't feel able to speak up and that makes me feel uncomfortable," he said.
"We need to have a system that acknowledges anyone can make an error."
Coroner Windley reserved her findings.
