I have been following Covid figures for the U.S., for Israel, and for Santa Clara County, where I live. In all three vaccination has been largely of the elderly, which one would expect to bring death rates down much faster than case rates. The only place where that seems to have happened is Israel. Even there, the effect does not seem to be as large as one would expect.
For each of the three I calculated the ratio between the most recent 7 day average of death rates that I could find and the earlier peak 7 day average, then calculated the corresponding ratio for case rates, assuming a 9 day lag between cases and deaths, and compared the two ratios. I chose a nine day lag because it let me match a death rate peak to a case rate peak. A longer lag might be more plausible, but I do not think it would change my results by much.
The U.S. and Israel give the death rate figures up to yesterday, while the county gives them only up to two and a half weeks ago on the theory that the more recent death figures are not reliable, so the interval for the county was shorter than for the countries. Since what I was comparing was not county to country but fall in death rate in the county to fall of case rate in the county and similarly for each country, that should not be a problem.
In Santa Clara County, case rates fell by a factor of 6.5 while death rates fell by a factor of 3.9.
For the U.S., case rates fell by a factor of 4 while death rates fell by a factor of 2.6
For Israel, case rates fell by a factor of 2.95 while death rates fell by a factor of 4.3
Only in Israel did death rates fall faster than case rates. I do not know if recent death rate figures for Israel are reliable, so redid the calculation using the same dates I used for Santa Clara County. That gave me ratios of 2.3 for cases, 2.6 for deaths. Deaths were still falling faster than cases, but not by very much.
Israel has had the highest vaccination rate of the three, so it makes sense that it would look better on the deaths vs cases comparison. But at this point, 11% of the U.S. population has been fully vaccinated, another 12% have received one dose. Vaccination has been largely, although not entirely, of the elderly. People 65 and over are about 20% of the population and about 80% of all deaths from Covid, so one would expect vaccination alone to have cut deaths roughly in half. For Israel, 80% of adults over 60 have received two doses of the vaccine, which should have cut deaths relative to cases more than in half, a larger effect than my calculations show.
I see two possible explanations for the pattern. One is that older people are not only much more likely to die if infected, they are also much more likely to show symptoms if infected; people who are infected but don’t show symptoms are unlikely to be tested and so don’t go into the count of cases. That would explain why deaths don’t fall faster than cases but not why, in two of my three areas, they fell substantially slower. And it requires that older people are not only more likely to show symptoms but as much more likely to show symptoms as to die. That does not seem to be the case, according to a source I found online.
The other possible explanation is that many of what are counted as deaths due to Covid are actually deaths while having Covid, people who die from some other cause but are tested and found to be infected. We would expect the number of those to be proportional to the number of cases.
If the first explanation is correct, figures on the number of cases overstate how fast it is falling, since symptomatic cases are falling faster than asymptomatic ones. Since asymptomatic cases are apparently still contagious, although less contagious than symptomatic cases, that implies that the risk of getting Covid from a random stranger has not fallen as fast as the decline in cases would imply.
If the second explanation is correct we have badly overestimated how deadly Covid is, hence probably over reacted to it.