You are assuming that hospitalization precedes testing, rather than positive testers self-hospitalizing upon diagnosis. Perhaps you are correct but I suspect it is more of the latter. If not, then mea culpa.
I’m not sure a hospital is going to allow someone to be hospitalized just because they test positive. You have to have symptoms that require hospitalization, to get admitted to a hospital. I don’t think they are counting ER visits, only cases where the patients are admitted.
Now, it may be the case that, because there are enough beds, doctors are admitting at-risk patients who test positive, even if they don’t currently have symptoms warranting, just so they can be watched, because the onset of severe distress happens so quickly.
The problem is that our statistics that we are using are useless. We don’t know what they actually mean, because we don’t have all of this information.