Hi Nicholas,
Thank you for posting back.
I understand the issue is that when you double click a .dotm template, it opens as the template itself with .dotm file format, it seems you cannot find Edit or Open option when right click the template.
I have tested in Current Channel of Word, version 2010(Build 13328.20356), I navigate to the template folder in File Explorer, when I double click a .dotm template, it opens as a new document which name is Document1 and file format
is .docx. I can find Edit and Open option with right-click, but only using Edit option will open the template in a new document.
Your Office version is a bit different from mine, to confirm the details and ensure we are testing in same Office, could you please share a screenshot as below?

Besides, it seems you modified the default personal templates location so the templates are not in
C:\Users[username]\Documents\Custom Office Templates
If it's convenient, just a test, you may try to copy one of the templates and paste to this location, check if same issue appears.
And you mentioned the issue starts after a Windows update. What's your current Windows version?
Best Regards,
Tina