After consistently receiving a "Staging Failed" error trying to publish the Export Web Map tool as a Web Tool, we can finally say we have succeeded in sharing a web tool to ArcGIS Enterprise 10.6 using ArcGIS Pro 2.1.
We checked this article about the "Staging Failed" error, but none of the listed causes seem to fit our issue.
https://support.esri.com/en/technical-article/000015108
The ultimate solution in ArcGIS Pro 2.1 was as follows:
a. Run "Export Web Map" Tool.
b. Share the results as a Web Tool. This is where we get the "Staging Failed" error.
c. Run "Stage Service" using the sddraft file from the "Share as Web Tool" step above. The draft file can be found under the current user name’s ArcGIS Pro sharing folder for the last process shared. (C:\Users\<USERID>\AppData\Local\ESRI\ArcGISPro\Staging\SharingProcesses\<JOBID>\<ToolName>.sddraft)
d. Run "Upload Service Definition" using the sd file generated from the Staging step above.
e. If not done as part of the publishing process, share the print service in Portal.
It might be helpful to note that according to this article, Web tools can only be shared with ArcGIS Enterprise server 10.6 or later ( http://pro.arcgis.com/en/pro-app/help/analysis/geoprocessing/share-analysis/what-is-a-web-tool.htm). Also, I believe ArCGIS Pro 2.1 is the first version to have the Export Web Map tool so maybe this will be fixed in future releases.
Incidentally, the only reason we used ArcGIS Pro instead of ArcMap 10.6, is that when we publish the Export Web Map tool with ArcMap 10.6, our web app builder print widget fails when we use that service. We traced this issue down to the print service published with ArcMap 10.6 having trouble with the print widget submitting "visibleLayers":[-1] when a layer has no visible layers in the map. It works fine on a print service published with ArcMap 10.3 but not 10.6.
I hope this helps.
--joe