Notice in that screenshot that my personal and workgroup (shared) template folders are what is shown, not the online Office ones, although I can click to get access to those. Generally I want my own templates. That can be accomplished with a registry hack.
Registry Hack for Backstage to Show Custom Templates by Default

You can also use the legacy FileNew dialog. A lot of people over the years have really liked this. It lets you divide templates into categories behind tabs if you want and lets you combine the listing of your own templates and any workgroup templates in a single dialog.
File New Variations in the Versions of Word

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Or, you can use custom software to give you template lists.

As far as I know, the Word object model (vba) has no control over pinning of documents or templates.