Statements made by individuals concerning the rise in death rates are being taken out of context to support anti-vaccine narratives.
The fact that death rates have dramatically increased from pre-pandemic levels does not mean that the vaccines are ineffective. Firstly, viruses are known for their ability to rapidly mutate and evolve, and as such it is not surprising that a 'new' virus such as COVID-19 has been able to infect fully-vaccinated people (such cases are known as "breakthrough infections") such as can be seen in the emergence of the Omicron variant.
This is consistent with data provided by the Centers for Disease Control and Prevention concerning the rates of hospitalization, in which the rate of hospitalization across the U.S. dropped following the vaccine rollout, and then picked up again as the Omicron emerged and began to spread.
Secondly, the data that Davison is commenting on is more complicated than anti-vaccine narratives are presenting it as. In the quotes being taken out of context, he is talking about an overall increase in death rates from pre-pandemic levels, and is not differentiating between deaths listed as being due to COVID-19 and deaths attributed to other causes. Davison implicitly acknowledges this by asserting the fact that even though COVID-19 may not be listed as the official cause of death on a person's death certificate, there is still the possibility that the official cause was triggered by complications from COVID-19. While this is true, this does not mean that vaccines are ineffective. Claims about the vaccines' supposed ineffectiveness based on present hospitalization and death rates fail to recognize that such an assertion could only be confirmed if there were a dataset featuring the same death rates drawn from a population that had not been given the vaccine.
As such, it is patterns in the data, rather than the bare numbers, that need to be paid attention to. As mentioned previously, the patterns suggest that the vaccines have been effective, and that increases in hospitalization ought to be attributed to factors such as inconsistent vaccine uptake and the emergence of new variants such as Omicron.