Not trying to, it spreads at a finite, measurable but rapid rate. It has a long incubation period, and a long duration until the virus is cleared. Most people remain infected for several weeks, with no or mild symptoms, but the small percentage who develop serious symptoms do not happen in a day, it takes time. Thus by the time you are aware you have a problem that is not just ordinary flu and pneumonia, it has already spread through a large part of the population.
You could model your hypothesis fairly easily. It is utterly inconsistent with the exponential growth that is actually seen. The exponential growth rate tells you the rate at which it is spreading. Adding in a time delay and a multiplier for fractional onset of serious symptoms only shifts the curve in time. It doesn’t change the exponential factor - which is the distinctive feature.