A web map does not save with editing enabled when the feature layer and web map are shared with a group in which members can update all content.
Last Published: August 6, 2015ArcGIS Online
Bug ID Number
July 29, 2015
August 21, 2020
Operating System Version
After review by the development team, it has been determined that this issue is related to a known limitation with the software that lies outside of Esri's control. The issue's Additional Information section may contain further explanation.
To have edit button show up in web map, go to the item details of service and explicitly say "Enable editing".