Near Me was originally released in the 1.12 version of Experience Builder. ArcGIS Enterprise 11.1 is running the 1.10 version of Experience Builder. I can't say for certain what kind of issues you may have trying to run a 1.13 widget in a 1.10 version environment. It may not be possible without extensive customization.
The first thing I would try is to open the manifest.json in Near Me and change the version and exbVersion to 1.10.0.
GIS Developer
City of Arlington, Texas