Regardless of the availability of testing at their disparate death investigation agencies, medical examiners and coroners across the country are guided by the National Vital Statistics System (NVSS) guidelines for death certification. A death certificate has two sections where the doctor who investigated the case will write the cause of death. Part I is the underlying disease or injury that starts the lethal sequence of events. Part II is for any other underlying conditions that the decedent had that made the death more likely.
The NVSS guidelines state, "If COVID-19 played a role in the death, this condition should be specified on the death certificate. In many cases, it is likely that it will be the underlying cause of death, as it can lead to various life-threatening conditions, such as pneumonia and acute respiratory distress syndrome (ARDS). In some cases, survival from COVID-19 can be complicated by pre-existing chronic conditions, especially those that result in diminished lung capacity, such as chronic obstructive pulmonary disease (COPD) or asthma. These medical conditions do not cause COVID-19, but can increase the risk of contracting a respiratory infection and death, so these conditions should be reported in Part II and not in Part I."
So, pathologists don't certify deaths as due to COVID-19 based solely on a positive nasopharyngeal swab. We get a clinical history of shortness of breath, chest pain, fever, cough. Yes, it is possible that someone could be an asymptomatic carrier and die of heart disease -- but in those cases we would certify the cause of death as heart disease and document the COVID-19 infection as a significant contributing condition, for several reasons.
Number one, COVID-19 can affect the heart (via myocarditis, pericarditis, or the formation of microthrombi). Number two, it's possible that the death may not have happened without the stress on medical resources caused by the pandemic. That's one of the reasons why the death toll in Italy is so bad -- their otherwise excellent healthcare system was grievously overloaded by a huge wave of COVID-19 patients. People who would've survived heart attacks during normal times died without medical intervention because they couldn't make it to the hospital or because the hospital couldn't treat them in time to save them. On some level it may be true that some natural-manner deaths being attributed to the virus could be seen as inflating the official COVID-19 numbers, but a failure to acknowledge and examine the pandemic's effect on the diagnosis and treatment of other natural deaths would also be problematic from a public health perspective.
To quote Dr. Ed Donoghue, a forensic pathology colleague at the Georgia Bureau of Investigation, "No matter how these deaths are currently being attributed, after this pandemic terminates, an excellent approximation of the true fatality rate of COVID-19 deaths can be made by the calculation of the excess mortality for the period. This calculation was very helpful during the 1995 Chicago heat wave. Almost certainly, because of the scarcity of testing and other reasons, we will find that the number of COVID-19 deaths has been grossly underestimated." The final death toll is going to depend on multiple factors: the density of the population; availability of testing; genetic factors (both host and virus); the public health response; and the robustness of the healthcare system.