We've found that the new map viewer will not add tables that are present in a map service to a new Web Map. The Map Viewer Classic will add them, but you are losing out on new functionality in the New Map Viewer. You can get the table into the New Map Viewer if the map was first saved in Classic, however their relationship to their parent layers is not present. We need layers and their related records to be fully functional in the New Map Viewer.
In speaking with Esri Support, they are aware of this limitation and asked us to post this idea. If this is an existing idea please direct us to it.