Hertz should be understood to mean "periodic events per second". I your case the events are the display of frames, so yes, you would be perfectly justified in using $\mathrm{Hz}$.
That said, as several commenters have already mentioned, the unit "Hertz" does not specify what kind of periodic behavior is being counted. So the author(s) or speaker must make that information explicit in some other way. The use of "frame per second" is one way to accomplish that: it is a more specific way to express yourself.
The real stinker with Hertz comes when you want to compare two closely related quantities like "angular frequency" and "frequency". The former is generally expressed in radians per second and the latter in revolutions or cycles per second. Transcribed into pure SI units both get written as $\mathrm{Hz}$ but they should not be treated as equivalent to one another. As a stop gap measure you can keep symbols for radians ($\mathrm{rad}$) and revolutions ($\mathrm{rev}$) around as if they were units, but they both represent pure numbers.
In the question text you actually pose a linguistic or philosophical question
To which (ontological) category does 'frames per second' belong, if any?
As I understand it, SI simply views this as the counting of something that happens periodically and as such "frames" is just a number without dimensionality.
There is no fundamental reason it must be that way, but the committee has decided to duck the question of what countable things should get their own units by saying, in effect, "none of them". Anything that you can count (in whole or fractional numbers) that happens periodically can have its rate denominated in Hertz.
An aside is that $\mathrm{Hz}$ is not the only SI unit equivalent to $\mathrm{s}^{-1}$. The becquerel ($\mathrm{Bq}$) also has that dimensionality, but should be understood to mean the (average) rate of some random or probabilistic behavior (such a radioactive decay).
