Completing death certificates is actually quite complex with regards to the sequencing of death and who can complete the certificate - the guidance on who can complete has been relaxed for Covid-19 as the preferred person may not be available/self-isolating etc.
People who complete death certificates are required to do so to the best of their "knowledge and belief" and use clinical judgement if no confirmed diagnosis, eg of Covid-19. If suspected, a box on the death certificate can be ticked to indicate this information may be available post-mortem.
The underlying cause of death, defined by the WHO as “a) the disease or injury which initiated the train of morbid events leading directly to death, or b) the circumstances of the accident or violence which produced the fatal injury".
There is also a part on the death certificate where other diseases, injuries or conditions that contributed to the death, but were not part of the direct sequence of death. These must be known to have or suspected to have contributed to the death - they should not be recorded if they did not. So the "died with not of" scenario should not apply.
So a Covid-19 patient may have the disease or condition which directly led to death as "pnuemonia" or "respiratory arrest" or something, but Covid-19 would be listed on the death certificate as a condition which led to the death. If the patient also had say cancer, it would be listed as something which led to the Covid-19 infection if it was believed this was the case. If the person had cancer and it was not thought to have led to the Covid-19 infection which led to the pneumonia that led to death, it would be listed as contributing to death but not related to the disease or condition causing it.
Add in to the mix that Covid-19 is a notifiable disease therefore must by law be reported to "the government". If a person was hit by a bus and found to have Covid-19, the infection would be reported but it should not appear on the death certificate as leading to or contributing to the sequence of events that led to death. Unless of course it was believed to have contributed e.g. the person was in an acute confusional state owing to infection and wandered into the road.
So to summarise - completing a death certificate is nowhere near as straightfroward as most would think and the process is not infalliable. There will be cases of over reporting and cases of under reporting. Reporting of deaths and reporting of cases of infection just show overall trends. They cannot categorically "prove" that every death has been correctly recorded.