Hospitals will be overwhelmed much more, and there will be far more deaths if we don't "flatten the curve." In addition, workers will stay home anyway (which many are already doing), collapsing the economy regardless.

It is indeed a false dichotomy. If anything, just irresponsibly letting nature run its course will be worse. Besides, recessions have been consistently associated with decreases of death rates. There is no evidence, and plenty of counter-evidence, that recessions cause increased death. Of course, we are getting a recession either way.