Sir Harry said that testing was the only way to discover how widespread the virus is, as he raised concerns about the "quality assurance" of tests being done at the drive-through centres.
At a virtual sitting of the Scottish Parliamen's new Covid-19 Committee, Sir Harry said he had been told "anecdotally" that there was more emphasis on quantity than quality of tests.
Asked if he believed all health and social care staff should be tested on a regular basis, Sir Harry said: "This might sound ridiculous but I think we should be testing the whole population. We need to know how widepsread the virus is in the population."
The Professor of Global Public Health at Strathclyde University told MSPs that the lockdown restrictions had forced isolation, which had worked in suppressing the virus, and gave his backing to the NHS tracing app.
"If we're going to move in a way that lifts lockdown and allows people to go about their businesses and meet their social needs then we need to know who is infected and who is at risk of spreading the virus to other people," he said. "There's been a huge emphasis on testing. People have set goals, and that's all very well as long as you know testing is accurate and who you're going to trace. There's no point testing if you're not going to trace and isolate.
"I've been hearing slightly worrying things about the testing process. A nurse has shared with me that when feeling unwell she went to a drive in place and the car window was opened and the person put in four testing kits and the nurse, her husband and children were told to swab their own throats. I'm not sure I would trust a 14-year-old to do that properly. The quality control around this - I'm not sure that it's there."
He added: "The fact that there's so much emphasis on numbers, is never mind the quality feel the width. We need to make sure the testing accurate, giving the correct information, and then there's follow up."
He added: "We need to impress upon the people doing the testing there's a way to do it proiperly. There are tests in development which focus on saliva so instead of a swab you spit into a test-tube which is easier to do - but the critical thing is to test those capable of spreading the virus.
"I hear stories of people with suspected Covid going to these primary care hubs and being told 'yeah you've probably got it just go home and stay in for two weeks', and they go home to a home with young people and others, and they've not been tested, so there's an inconsistency across the place - not just in Scotland.
"People who are suspected of having this virus all need to be tested properly."
He said the NHS tracing app was a "clever idea" and he "would have no hesitation in downloading it" and that it would be useful if enough people were encouraged to use it, but added: "We have to make sure there are people out there tracing contacts to isolate them and then you've got a more focused form of lockdown. So we've got a long way to go with test, trace and isolate.
"We need a clever, thought-through, process of transition and I don't think that's a sudden release of lockdown. We need to know what works and why it works in order to phase a way out of it."
Asked if it was time to move away from a suppression strategy to one of elimination, he said: "The whole idea of suppression is that it will lead to elimination. But you don't want small pockets of this virus hanging about and elimination will happen when you have a vaccine, which we're assuming is months away.
"I'm not sure we have an elimination mechanism just now beyond continuing to suppress and stop the virus. Viruses which kill lots of people tend to die out themselves very quickly because they've killed all the available hosts. This virus has killed a lot of people but it's still infecting folks which implies it's a very resilient virus.
"It's out there and lots of people are carrying it about without symptoms and passing it on without knowing. That's why testing is important, and to trace contacts and isolate them, that's the way you control the spread of any disease."
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