The fact that the template is on a network server doesn’t make any difference against a local template file as long as the OS hides the details of access protocol, which is the case of MacOS.
But Google Docs, AFAIK, is not a file server but a web application. It is known not to be fully compliant with ODF (the open document format) and I bet many features are not implemented or at least are not compatible with LO.
In a “standard” LO environment, templates contain styles and initial contents. After a modification of a template, styles in documents are updated only when documents are internally linked to the template (initial contents is never updated because this would destroy document edits applied after creation) and documents are then reformatted according to the new style definitions.
Now your question is probably about 1) how to update reliably, or 2) how to base automatically documents on a designated template.
Answer to 1: link document and template
Don’t double-click on a template icon. Indeed, you create a document derived from the template but the association with the template is lost and no automatic update is possible.
Create document with File>New>Templates and choose the template. Of course, this requires that the template is known by LO. This is done with File>Templates>Manage Template where you “import” your template. But this copies the template into some internal LO location which means you must reimport after any change to the template.
You can make your templates automatically known to LO by adding your own template directory in Tools>Options (probably Preferences > something under MacOS), LibreOffice>Paths.
Answer to 2: new documents associated with a template
To simplify your life, a template can be designated as the default template in File>Templates>Manage Templates so that it is selected whenever you File>New>Text Document or Ctrl+N (⌘+N).
