But John Apter, chairman of the Police Federation, which represents rank-and-file officers, said Ms Billingham's frank assessment "reflects the reality of what we are experiencing every day".
South Yorkshire Police has been praised for the way call handlers deal with people with mental health problems.
HM Inspector of Constabulary Zoë Billingham told the BBC that police are clearly doing a good job and definitely have a role to play, but said they are "over-stretched" and "overwhelmed" and the current system is "failing people".
HM Inspector of Constabulary Zoe Billingham said the inspection found police respond to people with mental health problems with care and compassion.
And Police and Crime Commissioner for Wiltshire, Angus Macpherson, said those with mental health problems need "specialist support and care", rather than being left to be dealt with by police.
The top five individual repeat callers to the Metropolitan Police all have mental health problems and called the force a total of 8,655 times in 2017, the report said. It cost Scotland Yard around £14,000 just to answer the calls of a single person who rang 999 more than 1,000 times in the space of a year.
"IT IS frustrating" that mental health demand on North Yorkshire Police can not be accurately measured, the Police, Fire and Crime Commissioner has said.
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"As a Force we have been vocal about changing the way people with mental health issues are treated".
The findings come amid intense scrutiny of police funding and performance.
The chancellor of the exchequer unveiled plans for a new NHS mental health crisis service in the Autumn budget this year, which is set to include mental health support in every major A&E department, more mental health ambulances, "safe havens" in communities and a 24-hour mental health crisis hotline.
The report pointed out that South Yorkshire Police's call centre supervisors carry out regular performance meetings with police officers and staff.
The report, carried out by Her Majesty's Inspectorate of Constabulary and Fire and Rescue Services, makes it clear that whilst the police service is doing "a good job in hard circumstances" there are concerns over whether forces should be involved in responding to mental health related issues "at the current level".
"It is right that the police are there to protect those in immediate danger, but they shouldn't become the first point of call for those who need longer term mental health support and access to prevention measures".