I tried the workaround in this related issue: Upgrade Portal from 10.5.1 to 10.6 fails (giving the service account full access to the C:\Program Files\ArcGIS\Portal). That did not work either.
It was probably a nuclear option, but we:
- Uninstalled Portal on the server
- Deleted all existing Esri directories
- Reinstalled 10.5.1
- Restored from a 10.5.1 backup
- Then upgraded to 10.6.1 successfully
In retrospect, it was likely related to a portal database locking issue (the new database was having trouble pulling data from the old database). It would be interesting to check the portal Postgres logs to see if a connection attempt is failing. If so, you could probably add a line to the conf file to accept the connection.