If for example you are in a still strongly growing (but no longer exponential) regime where you can approximate the currently infected number of persons as proportional to t^2 , and there is some constant IFR, then f'(t) is also proportional to t^2. f(t) will then be growing as t^3, so f/f' will go as t^-1, which will look quite similar to an exponential decay at the timescale of the process.
You may see some wobble caused by outbreaks here and there, but the mechanics of COVID-19 are always the same.
Going to zero exponentially does mean it will converge mathematically. And t^-1 won't look like the things that can be observed. There is no such thing as a constant upward trend away from the exponential. If you look carefully and do some regressions you can see some ups and downs. Changes in the behaviour of the people are able to modify the trend, but there is no constant movement away from an exponential.
You can always add an upper exponential bound that will converge below single digits per day before there is any chance that the lines cross.
I'm not claiming that is what you were saying, I just wanted to call out this point in case other people read this that may come to this conclusion.
FWIW, the solution to f'/f = exp(-t) is f(t)=c*exp(-exp(t)), which actually looks like an epidemiological curve, but like any sigmoid it will be hard to determine the actual saturation point until after it has already passed the inflection point.
