The difficult thing about measuring causes of death is that we take it as axiomatic (and, to be clear, I think we're right to do so) that the goal is not for humans to live forever, that everyone should die of
something. Heart disease mostly affects people who are at the age that people tend to die of "old age." In other words, eliminating heart disease entirely would largely only cause those people to die of something else. There are, I don't know, maybe 100-200K "premature" deaths from heart disease which ought to be the focus of public health interventions, but addressing the rest of them won't meaningfully save lives.
Or put another way: 22% of all deaths in the US happen in nursing homes. But we don't say we should get rid of nursing homes. If 22% of all deaths in a city happened in their college dormitories, there would be an immediate outcry to shut down the dormitories.
Now, the harder question is that covid also disproportionately affects people near the end of their natural life (and probably so does almost every medical cause of death besides, like, SIDS and pregnancy).
So the question is then, how many people does covid kill "before their time," so to speak, and how does that compare to heart disease? One of the common arguments is that it does kill lots of people before their time - even thought it skews towards the older, it nonetheless kills a good chunk of older folks who would have otherwise lived happy and healthy lives for several more years.
One way to measure this is by looking at excess deaths. How many people are dying of any cause compared to the year before covid was around? If covid was, say, killing people who were about to be killed by heart disease, we'd kind of expect a drop in heart disease death statistics. If we're not seeing that, then that implies that covid shouldn't be bucketed as an "old age" cause of death, that there's a reason to be more concerned about the 630,000 covid deaths over the past almost two years than the 659,000 heart disease deaths in 2019. And in fact it looks like the number of heart disease deaths didn't drop in 2020 (and grew, in fact) even as covid killed 350,000 people.
(You can get these numbers out of http://wonder.cdc.gov/ucd-icd10.html - which I found linked from the site you linked, thanks! - and break them down further by age etc.)
A lot of arguments that we should specifically care about covid (which is the camp I'm in) are predicated on the assumption that this is true; a lot of arguments that we should just treat it like we treat the flu are predicated on the assumption that it is not. I think this is a question of fact (though one that requires the statistics to actually be available) and not just a difference of opinion or worldview.