Zoning too much at a time is probably the primary cause of the problematic death waves, and to some extent it's a reasonable (realistic) enough result from a simulator, given the initial factors: If in the real world you had a very large number of people, roughly the same age, moving into a new area at the same time, statistically many of them would die around the same time, causing a number of problems. That makes for somewhat fascinating emergent behavior in a simulator. The compressed timeframe in which the game runs makes the effect all the more pronounced. Once the death spike goes beyond what your current deathcare can handle, it starts to resonate with abandonment and new influx, adding to the influx after the deaths themselves, thus both increasing and sharpening the influx spikes. The death waves then begin to run in a cycle, becoming increasingly problematic unless you do something to mitigate the problem. For instance, you can manually force people out of the city to spread the influx e.g. by cranking up the taxes until a desirable number of people have gone. Timing is essential, but this can be used to combat the massive spikes.
Fascinating as it may be, there is gameplay to be considered here; some things should be done to reduce the likelihood of these events spiraling out of control, particularly when it isn't clear to the player what caused it: Did they or did they not realize they might be zoning too much at a time? With a larger spread in age among people moving in, and perhaps a larger spread in the age at which death occurs, I believe you would increase the chances of the deathcare services being able to handle the deaths when they happen (as long as you do have reasonable coverage). The inner workings of deathcare could perhaps use some improvements too. But the game only has so much tolerance for heavy zoning (population influx spikes), and that is a fundamental and real problem that cannot be avoided in a simulator such as this, unless people moving in have their ages spread across the entire relevant spectrum. The player may need to develop some understanding of this, simply as one of the realities of a city simulator – along with traffic jams.
That said, I'm under the impression that the developers are acutely aware of the problem of what makes good gameplay, and I have faith this will be polished out in time.