Our server is version 10.3. I tried replacing the FeatureServer URL with the Map Server URL, and the table became usable. But I don't like using the MapServer URL because I don't have as much control of the layers within the map.
Interestingly, I created an app with WAB and looked at the web map through it. When viewed through the app, the table is completely visible. So that kind of fixes the ultimate issue, but I would still love to know whey the table isn't visible in the web map in the map viewer.