Does this issue only occur to this specific Outlook 2013 client? Please check if other users who use Outlook 2013 as well have the same issue.
doing Choose Form and User Templates in File System doesn't show the list of template files to choose from.
Where did you store your email template? In the default location like C:\Users\Administrator\AppData\Roaming\Microsoft\Templates? If so, to conduct further researches on your issue, it is suggested to copy the same template to other custom path and see if the issue has any difference. Also, in order to rule out if the issue is related to the Templates folder, you could try to rename the Templates folder as Templates1, restart your Outlook and copy the template file to the new folder to see if the issue will happen again.
There could be some add-ins in your Outlook client that may cause this issue, so it is suggested to start your Outlook in safe mode (Press Win + R, type “outlook /safe”, press Enter.) and see if the issue continues.
In addition, the use of template files by antivirus software or other programs may also cause similar problems, so when the issue occurs, please try to exit all other programs to see if it works.