The physics is that the light cone - which is what these plots show - is a hyperbola. I provided references in another thread. But if fitting to a parabola is easier then it won’t do much harm.
The main thing I’d take away from this analysis is that reporting the HFR as zero for images where no stars are detected is not a good idea. Reporting a high HFR or, better, not using that point would be more useful.
The second of the failed example plots doesn’t show anything that can be used to determine focus, one point with a high HFR with the rest showing more or less the same HFR varying apparently randomly.
The last failed graph shows a situation which is part of the problem, the SGP HFR detection falls apart if slightly out of focus. In this case the left hand point should have a HFR of about 9 but it’s 3.2. The first failed graph also shows this where the two zero points should be about 13 and 15.
I think the essential is to detect that the HFR determination is giving bad data and to manage this in a better way.
But what I think is really needed is to be able to get star sizes reliably even when well out of focus. We can see star discs or doughnuts when a long way out of focus so there’s a prospect that a different algorithm could do this.
As for using the variance in HFR as a measure of quality I think that the main influence on HFR will be star brightness. Bright stars will be larger. Experimenting will be needed, is the data on star brightness, HFR and ideally position in the frame for each star available?