Robert,
I got confused about your second answer, I used the map tab but seems that I cannot add any service published by ArcGIS Server, all the available options are for services from AGOL (please see the first screenshot).
In addition, I could not find any widget to add services (please see the second screenshot, do you mean the “map” tab in the first screenshots?)
Please provide me with a YES/NO answer to this question: Can I add a service published from ArcGIS Server to an app built with WebApp Builder?