How to approximate the peak of an epidemic (depending on epidemiological parameters)?

You can always make some reasonable approximations which simplify the problem for certain special cases. For example, you can assume that the infection spreads very quickly. Then, at the beginning, nearly nobody is infected, i.e.$S/Mapprox 1$, while towards the end, nearly everybody is infected (or recovered), i.e. $S/Mapprox 0$. If everything happens quickly enough, you can ignore the time in between, i.e. get a reasonable approximation by assuming that the switch between nearly nobody being infected and nearly everybody being infected is nearly instantaneous.

The approximation $S/Mapprox 1$ for the beginning results in an ODE system which is linear and for which you can thus simply write down the solution. You then determine for which time $T$, $S(T)=0$. At this time, you then assume $S/M=0$, which again results in a linear ODE system which you initialize with the state of the first one at $t=T$. You then set $I(t)$ to the corresponding solution of the first ODE system for $t<T$ and to the solution of the second system for $t>T$.

PS: surprisingly, the approximations seem to become even better if you assume a fixed time delay between $I$ and $E$, i.e. if you set $E(t)=I(t-tau)$ with $tau$ the incubation time. Depending on the case, this might be even more precise than the exponential distribution you currently assume. Alternatively, you might also assume a mixture of both to e.g. say that there incubation period is at least 3 days and from there on follows an exponential distri. The point is that an explicit time delay somewhat decouples the ODEs. Essentially, this means that $S$ and $E$ are "living" at time $t$ and everything else at time $t-tau$. If you now manage to quickly deplete most of the susceptible population in a big final wave of infections lasting for $tau$ or less, the concentration of $I$ will just rise exponentially troughout this final wave even though $S$ will already quickly decrease (essentially, $I$ only "realizes" that a significant part of the population is already infected and thus that $S$ is already quickly decreasing with a time delay $tau$). Once you are through the final wave, $I$ will start to stop growing exponentially and do complicated things, but this doesn't matter anymore because everybody is already infected.

In short: what seems to make the problem more complicated (adding explicit time delays) actually makes it easier to find good analytic approximations. This is thus one of the few occasions where being more precise and realistic is simplifying the calculations.