If we're using that logic then there are also 9 Merlin engines on each Falcon 9 and 3 RS-25's on each shuttle.

10 per mission, don't forget the 2nd stage. And 28 per FH mission.

You might be mis-remembering a bit. The list of Saturn V launches at Wikipedia shows some anomalies, most notably Apollo 6, where two second stage engines cut out early -- but those were J-2 engines, not F-1s. It doesn't show any F-1 anomalies, though the F-1s never reached orbit in any event; as first-stage engines, they were shut down and jettisoned long before that even on a completely normal launch.

https://en.wikipedia.org/wiki/Saturn_V#Saturn_V_vehicles_and...

Ah, yes. I was remembering a center J-2 on the second stage of Apollo 13. So, my mistake.

I think it's fair to count the number of missions (it's at least consistent across engine types), though I'd probably prefer to also see the number of engine-missions as well.

But in defense of counting the number of missions, each mission has similar "external" characteristics: ambient temperature, sunlight level, burn duration, wind shear, humidity, etc. A combination of these environmental characteristics could cause a failure (Challenger failure was caused by excess cold). Since the environmental characteristics are the same for the mission, the number of distinct missions an engine has performed is a decent proxy for robust performance throughout the space of environmental conditions.

That would make more sense if, in fact, engines failed or didn't fail as a group, but in fact they tend to fail individually. This can make the difference between success and failure for the mission, when other engines of the same type fill in for the failed unit.