The two negatives were PCR tests. Those are different from antigen tests. The blog post you've linked says there is a problem with false positives with PCR tests, but that doesn't seem to be relevant here, given that both of those results are negative.
So this could illustrate some issues that people should bear in mind when considering the antigen policy.
Apparently viral load ramps up really fast, like hours, until the immune response kicks in, and it is at that high titre that you are most infectious. If everyone tested everyday, and isolated on a positive, then this could all be over by Christmas.
My understanding was actually that antigen tests are far less sensitive relative to RT-PCR but have high specificity. If that's correct, then DeWine's false negative would be a relatively rare event. By the way this is an excellent podcast episode covering the topic: https://www.microbe.tv/twiv/twiv-640/
Thus there was/is an emphasis on positive test rates rather than number of positive tests. However this distinction probably has been hard to get across to the public at large.