ArcGIS Online does not recognize a printing service if it is not in the default printing directory:
Last Published: April 20, 2015ArcGIS Online
Bug ID Number
March 31, 2015
August 21, 2020
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.
The an executionType must be esriExecutionTypeSynchronous, not esriExecutionTypeAsynchronous
Only publish the default print service if it is going to be used in ArcGIS Online.