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.
I agree completely with the math, you are missing the point. Of course the model for infection is well known, the problem is to fit a model to an actual event you need accurate data. The data in this case is terrible. We are only testing a small fraction of the people infected. I have been exposed and have had moderate symptoms worse than a flu, but not bad enough to need critical care, same for tens of thousands of others, but we can’t be tested due to rules and lack of test kits. Likely a hundred thousand others are infected with no symptoms, or symptoms too mild to bother with. So when we see 1731 new cases added tomorrow and 2132 the next day, it is meaningless if the actual numbers are 27,321 tomorrow and 21,238 the next day. We have such a small, biased (those sick enough to be hospitalized are the ONLY ones being tested in my area) sample that it doesn’t really provide a sufficient data set to model, all we are modeling is the limits of our testing capacity.