Also your 50% seems very high to me. My understanding is that 50% is about the death rate of covid patients requiring ventilation, so the overall death rate of people hospitalised for covid must be well below that.
Also if you factor this timing impact, you have the same timing impact on the denominator, i.e. people who are infected but haven't developped antibodies yet.
 in fact for the 50%, we have an idea from Cuomo's presentation charts. The gross daily hostpitalisation rate as of 10-15 days ago was about 2000-2500 per day, and the number of death now is about 500 per day, so that suggests roughly a 20-25% death rate for hospitalised people (I am sure there must be some more precise figures somewhere).