You're totally right, of course. And I even mention that in the post. If we assume that S ~ N then we get that exponential growth. But eventually (unfortunately) enough people will get sick that S is no longer close to N; that's why the epidemic peaks.
In this model, change in the number of people is function of the number of people infect, that's it. The number of potentially infectable people is so large that changes in it's value won't enter into consideration until the US is at WWII-invasion level of fatalities.
dS/dt = Sr, for some value of r.
A simple differential equation indeed, solved by S = e^(t*R); exponential growth as we see (note, I'm not claiming my r here is R0 or whatever).