an overwhelmed healthcare system is also more likely to detect fewer of the infections, so the denominator in the fatality rate (the number of cases) will be lower than the real number of infections.

Especially in eg Italy with the case count still increasing rapidly (likely indicating that most of the increase is from detection catching up with previously infected more severe cases, rather than from detecting new infections) it seems likely there are many more infected but asymptomatic or low severity people who will not become a case.

It’s also pretty obvious that when a country reports deaths on the very same day they report their first cases (as both Italy and Iran did) that they will likely have had uncontrolled community transmission for perhaps a week or more before that date, putting detection (and thus number of cases and the denominator of the fatality rate) behind the curve.

Deaths almost always get counted, and usually on the day they occur. Infections on the other hand are much harder to count accurately, and when things are overwhelmed only the most serious ones will get counted.