I don't see what is wrong with not counting positive tests twice over in the the daily count. For example, if I test positive Monday, and then go get tested Friday, the other methods would count me as a positive case twice. If anything, that inflates numbers in a way to me. I am not two separate people who contracted the virus.
The different ways to measure positivity give you different slices of information. It's good and honest and transparent to put out all three. So far, all three have pretty much agreed on the overall trends (see chart above).
It's disingenuous to switch to the one that fits one's narrative best and claim it's the right one and ignore the other methods.
And, as I've said several times: positivity can misleading if you're only testing the people who chose to get tested. A large number of people worried about having COVID but having no symptoms will drive positivity down. OTOH, if the vast majority of people getting tested are doing so because they have symptoms, then the rate shoots up.
Only randomized sampling will give a true snapshot. (And before anyone asks, you entice the people you're targeting with a reward or some type of coupon since it's voluntary to get tested.)