I think I made the same mistake you did this past winter when calculating the death rate of hospitalized patients. Namely, there is a difference between newly-admitted to the hospital and net hospitalizations, because there are two ways to leave the hospital. Unfortunately I don't know whether there is a dataset that contains newly admitted rather than net hospitalized because the former is always larger than the latter and thus underestimates the denominator in the death rate.
If you subtract from day to day the total number of hospitalized you will not get newly hospitalized number, because some of them would have left due to getting healthy, and that lowers the death rate, and we have no idea by how much.