I'm curious about your data set. The correlation of cases to deaths using data from covidtracking.com for the US over the past year is 0.28-0.3 when I ran it. I slid it by two, three and four weeks.
* A peak of cases around Apr 11, followed by a peak of hospitalizations on Apr 22, with a peak of deaths also around Apr 22.
* A peak of cases around Jul 22, followed by a peak of hospitalizations around July 26, followed by a peak of deaths around August 4.
If I were going to do a more detailed analysis, I would want to try breaking out individual states/counties (subject to some reasonable population minimum), such that multiple distinct trends nationally don't interfere with each other in the data.
I really wish that stochastic testing were discussed more seriously.