Tim,
I don't believe the false positives are evenly distributed across society. When the sample is taken in a clinical setting at the time someone is showing symptoms I feel the false positive rate is more likely to be where the ONS and PHE say it is.
I think the majority of false positives, or "symptomless cases" are amongst the mass of the public who are advised to take a test for whatever reason at home or a walk/drive in centre.
So, 117,588 people have died with COVID mentioned on the death certificate. Some of these deaths are so recorded because a death reported as with covid needs no further investigation, reducing workload in the hospital system. I think that we should deduct the recorded deaths from the total number of "positive cases" and say that of the 3.8 million positives the number that really had COVID, recording real symptoms, is probably less than 2 millions.
We will never know the real figures, there will be endless postmortems and scientific publications over the next few years, but for most of us the bottom line is get vaccinated, and then get back to normal life.