It can’t be done effectively during exponential growth at the current rate: there simply aren’t enough identified recovered patients to supply serum to the currently sick patients. If the exponential rate decreases a bit, maybe.

FWIW, it seems at least plausible to me that a recovered patient who was treated with serum may not generate as many protective antibodies as a naturally recovered patient. For example, Rh antibodies are used during pregnancy specifically to prevent an immune response. I don’t know if this would apply to COVID-19 or if giving serum only after symptoms become severe would prevent this outcome.