Errors in which sense? False positives like here, which can then be resolved with the better PCR test, are much less of a concern than false negatives.
In the current situation where it may take 10+ days to get a lab test (for non-governors), it is an issue. For a warehouse worker making minimum wage, 10 days of missed work could lead to financial ruin. If we get to a point where most screening is done via abundant cheap antigen tests and RT-PCR capacity is available for confirmation then this would be less of an issue.
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/
The concern is whether the elevated positive tests that we are seeing is due to false positives. This is probably the case if it is the antigen tests that were administered.
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.
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/