Here's another issue you can investigate to see if it might be the problem:
For More Information see: Can't login to FMS database on new Mac, other Macs can
This is one of many acknowledged bugs that can be found in the Known Bug List thread here in the Report an Issue section of the forum.
It can also be downloaded as a database file from: https://www.dropbox.com/s/jt09b82i0xijbu3/FMP%20Bugs.zip
It worked! I copied the server.pem file from another Mac to this one and my remote database is now visible and opens.
I wonder what causes FileMaker Pro to incorrectly skip this file during installation?
Thanks for your help!