Yes, most covid deaths occurred in those with pre-existing diseases. But the primary doctor judged that covid (probably including the pre-existing disease) had played a role in the death, thus writing covid as part of the cause of death in the death certificate. I think this article explains this well:

https://www.aamc.org/news-insights/how-are-covid-19-deaths-counted-it-s-complicated

There are numerous reasons why we determine covid as the cause of death. For example, the hyperinflammation responses against covid can harm various organ, including those burdened with pre-existing disease (this don't occur with vaccines). Another reason is that covid autopsy has found covid as a convincing cause of death (or the final bullet), on top of pre-existing diseases (such as finding evidence of SARS-CoV-2 RNA/proteins co-localizing with damaged tissues or additional covid-specific pneumonia/blood clots).

In the end, I think determining the cause of death is up to the primary doctor, because most death cases are not autopsied. And some doctors may have different judgement than others.