I don't see any other way to track testing data. There's no reason to count someone twice when they're only 1 person, and the point of doing lots of tests is to find it before it gets out of hand, I don't see the issue with counting someone as negative multiple times.
The governor said the state can re-open when the positivity rate was below 10%.
Person A tests positive
Person B tests 10 times, all negative
Rebekah Jones would say the positivity rate is 50%.
1 positive test/ 2 people tested
Her supervisors would say the positivity rate is 9%.
1 positive test/ 11 tests total
Above is for illustration. IIRC their calculations were something like 12% vs 8%, when the governor said it's okay now to re-open. Do you see a problem with that?