well without going through every single death certificate and being able to determine what level of cycle threshold was used in each deceased individual then I would look at the false positive rate of the PCR test which is between 0.8% and 4%. I would then subtract that false positive rate from the total number of tests undertaken,not from the positive tests recorded.Very important. I would then be left with a figure of false positive tests that I could subtract from the number of positive tests recorded and it would leave me with an approximate figure of true deaths.
Okay, so first off, we're carrying out about 20 tests per case detected, which is to say 5% of them are positive, so for that 4% figure to be true then you have to think that 80% or so of the positive tests are in fact false positives, likewise back in June/July we were doing 200+ tests per case detected, so even 0.8% false positive looks unlikely when only 0.5% of tests are positive anyway.
The 0.8% - 4% figure comes from Mayers & Baker (3 June), that paper starts with the words
"the UK operational false positive rate is unknown"; as far as I can see the specificity of the test is now estimated to be 99.9% or higher, which is to say <0.1% false positive, so the test itself is highly accurate, the only substantial risk comes from cross-contamination and as evidenced above there doesn't seem to be much of that.
Next you will want to consider the fact that the probability of a test being a false positive is dependant on the probability of the patient having the virus in the first place, across the summer with very little virus about the chance is quite high that any individual test is false, in the middle of winter with lots of cases the chance becomes much smaller.
You will of course be aware that tests are usually only available for those with symptoms, thus people who take the test are likely to have the virus, and the chance of a test being a false positive becomes vanishingly small.
You'll also want to remember that those who've died after a positive test will likely have shown symptoms of the virus first, and died of something afterwards, which is a clue that something might've been wrong with them and again further reduces the chance of that test being a false positive.
None of which is to say that the 28 day measure is perfect, some deaths will be unconnected to the virus, not all deaths with a positive test end up with Covid on the death certificate, on the up side it is the most rapidly available measure so it's a good way to see which way we are going.
Finally, there is the simple fact of excess deaths, which doesn't care about PCR tests, or what's on the death certificate, and which also shows over 70,000 excess deaths so far this year.