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.
In what way is the fact that there will be two different levels of economic costs in two scenarios a 'false dichotomy'?
Edit: I think this is worth clarifying. I may misunderstand you, but it looks like there are two errors in your post. The first is treating differing scenarios with differing outcomes as if they were the same, simply because they have some facet in common.
The second is a mis-use of the term 'false dichotomy'. If you believe that a choice between two options will have the same outcome, that is not a false dichotomy. A false dichotomy is a logical fallacy, a special case of the fallacy of exhaustive hypothesis. Here, one wrongly assumes that there are only two possible choices and uses this assumption to arrive at some seemingly logically derived conclusion.
My apologies if I've misunderstood your intended meaning from your short post.