No, because the virus takes time. Survivors aren't "survivors" until they've recovered, so they reflect the people who "would have been" critical cases maybe 10-15 days ago. But 10-15 days ago the number of cases was (depending on which doubling statistic you use) 4x-32x smaller than what it is today.
Exponential growth messes everything up, basically. All our intuition about how things work tends to be wrong in subtle ways.
e^(ct)/e^(c(t - 14days)) = e^(c*14days) = k = constant, with c the growth rate. While you are right that, in an exponential growth scenario, the number of recovered cases is smaller than the number of currently active cases, the factor is proportional to the growth rate which is worst case constant, but likely going down over time due to quarantine measures. As such we have theoretically enough recovered cases if n_infected/n_critical > k. For the most critical age group an estimated 10% of infected require ICU care [1]. Thus we need k < 10. With a 1:2 ratio for serum we get k < 20. Likely not everyone will donate though and constant factors in this equation will end up mattering but the exponential nature does not kill the idea in the manner you indicate. However, it still requires scaling production exponentially. This is true for all other drugs as well though.
Exponential growth messes everything up, basically. All our intuition about how things work tends to be wrong in subtle ways.