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/