If we expect a roughly fixed number of people to eventually need hospitalization, then underutilizing hospital capacity is a waste of very expensive shutdown time.

The shutdown is preventing quite a bit of sickness, disability and death from occurring during the course of the shutdown. If we could indeed maintain a shutdown forever, and if that would reduce the total amount of sickness, death, etc, then maybe it follows that an indefinite shutdown would have value to weigh against the cost. But those are some seriously questionable assumptions. What actually seems to be happening is that we are delaying people getting sick, and the value of that is not self-evident.

I am suspicious that much of the COVID-19 response is based on pretending that it is a severe pandemic flu with a comparable fatality rate. With a flu, shutting the world down until summer or until a new flu vaccine can be mass produced has clear value. But COVID-19 is not the flu.