The issue came from me trying to take a shortcut (shame on me...): I had another map w/ our editable layers in it. I saved it as a new map, then used AGIS Online Assistant to re-point the layers to the publicly-shared views of their editable counterparts. AGOL Assistant successfully remapped the layers to the public views, but left some remnants of the previous editable and not publicly shared layers in the map's JSON settings. Those remnants must be trigger AGOL to ask for a login, which explains why once you cancel out of the request the app works as expected. The problem was solved by creating a new map, bringing in the publicly shared layers to start with, and pointing all the dynamic text, etc in Exp Builder to the new map and fields.
