This is against my better judgement, and it's sort of pointless, but I can't bring myself to not respond. :/
Nothing "technically invalid"? It's nonsense.
It's nonsense in the sense that it doesn't really
embody what people mean when they talk about averages, but it's a perfectly clearly-defined usage of the concept of "mean." Obviously the mean of a single sample doesn't offer a ton of confidence in determining population means, but as ultra-simplified cases in mathematics go, if anything it's relatively non-degenerate.
How so?
A frame of source video is, in a way, a sample entity allowing me to see what happened during a 60th of a second. If I see one new frame of game output in a frame of source footage, then I have (1 frame)/(1/60 s). If I rewrite this to separate the coefficient from the bottom unit, this is 60fps. Similarly, if I see 0 new frames of game output in a frame of source footage, that's 0fps. This extremely short sample period makes the sample values heavily quantized, but since it's quantized in a way which doesn't round "partial frames" away (rather, it counts them "as members of preceding/following frames"), it shouldn't apply a measurement bias and should be simply an imprecision that we can mitigate via averaging over multiple "samples".
Let's suppose we sample a video file and start seeing a pattern like 1 0 1 0 1 0 1 0. Our framerate interval samples thus look like 60fps 0fps 60fps 0fps 60fps 0fps 60fps 0fps.
If you start averaging those, you'll find it's pretty clearly converging to 30fps, which is what you'd intuitively expect from the every-other behavior. If you average them over a whole second, the result you get will be equivalent to the "count the number of frames that showed up in that second" method.
FPS means "frames per second", which is merely a rate, not "number of frames counted during a 1-second sampling interval." Requiring 1 second (60 frames of a 60fps source video) of accumulation to comprise "a sample", as you have done, is entirely arbitrary as a choice of sampling interval; there's nothing intrinsically natural about using 1 second. Does it produce a meaningful "sample" that you can work with using methods of statistical analysis? Absolutely, but it's it's
also an average of values sampled over smaller time-scales (which can themselves also be worked with using the same methods of statistical analysis). The only thing distinguishing "1 second" from other intervals is that we've picked our units such that the denomenator of the frames/time fraction has a coefficient of 1, which makes the division "invisible" (and resultantly, the framerate measurement is "just a count" instead of "a count followed by a division operation").