Can a "supervisor" of an APFT also serve as the "scorer" of an APFT?

Short answer, No. With the 4 person minimum the NCOIC, Supervisor, Scorer and other person to hold feet there should not be an issue. If the other person holding the feet is another participant than so be it but the 4 person "rule" still applies and there should not be a double up of any responsibility. That is how I would interpret.

Is the APFT being graded fairly for everyone? If so, I wouldn't sweat it. What's more important, an accurate APFT being conducted or that the regs were followed to the letter?

In small units it is difficult to give everyone just one additional duty. Some guys just need to be double hatted. If things aren’t being done right or fair then maybe the solution is to consolidate elements so the supervisor isn’t grading. I would keep away from situations like you described because people will question the integrity of the test.

Retired 93 - In my day it could be done but not, 'For The Record".
I gave many assessment APFTs to soldiers that were weak in the test or a particular event. I also did it for assessments of overweight soldiers, soldiers that were returning to full duty (seeing if they were really 'Fit to Fight') plus often for competition between teams, squads, sections and platoons.
None were for 'For the Record'. Some were even for fun.

So just some clarification. 7-22 strait up says in black and white that for the "alternate for record PT events (swim, bike, or walk) that a supervisor WILL NOT be a scorer. This exact phrase is missing in the section talking about a regular APFT (though it was in the previous reg 7-22 replaced which was 21-20).
The gray area stems from the fact 7-22 clearly says you need a minimum of 4 people for an APFT (a NCOIC, supervisor, scorer and someone to hold feet for sit ups). It goes on to say that in a situation of short manning test takers can hold each others feet and "NCOICs can serve as the Supervisor." So again it NEVER says supervisors specifically CAN be a scorer even in low manning. But it also never specifically states the CANNOT.
The problem is we have guys grumbling; The point of having a separate Supervisor and Grader in reg is stated so the Supervisor can assist the Scorer in making calls on form and count." So in other words a second opinion. If the Supervisor IS the scorer; then obviously there is no check on anyone and no second input.

I wouldn't see an issue with it if there was another APFT Supervisor while the current one was grading. We always have 2 supervisors and one will grade if really needed which will leave 1 supervisor to observe and control.

In the IT community, we call it separation of duties. Something has to be sacrificed if its one person doing both; with accuracy usually being the price. So there is an upside to someone looking to cheat, especially in a big unit or flight.