I am not sure where the disconnect is here.
Maybe I am not doing a good enough job of explaining what I am saying?
We are not identifying all cases. Initially we were only testing people with symptoms.
Of course the rate of hospitalizations would be higher then because you are excluding people with no or mild symptoms.
As you expand testing, the rate people will be hospitalized will go down.
This sounds downright Trumpian.
The hospitalization rate is hospitalizations divided by positives, not total tested.