i don't see the idea that he had it for a while and was getting over it when he popped positive as being consistent with that testing pattern,
The difference is this. When you recover from the virus, (inactive) traces of it can stay with you for a long time -- weeks. Those will be detected at a much lower rate than when you are infected. When it is detected as such, it's considered a false positive (because you don't really have the virus) and (I think) those comprise the majority of false positives.
Anyway, my point is this. I think that after a positive test, for a player with no symptoms, he should be given an antibody test. If that's positive, he's good to go. Antibody test results can be same-day.