I suppose this is the expected behaviour.
Whenever you open a template, the only location information Writer knows of is the templates folder. Obviously, you don’t want to save your new document there.
New command clearly requests the creation of a new document which is temporarily name Untitled n. The first
Save command will behave exacty like
Save As to offer the possibility to change this default name. As @anon73440385 mentions, the file picker starts at the My Documents path in the options (user profile) or the folder used in the last call to the file picker.
In may OSes, the file browser shows a list of preferred locations at left of dialog. Mine (Linux) shows Home, Dektop, Documents, Downloads plus a list of devices. The list can be edited with a right-click to add other preferred locations.
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Additional note 2020-02-29
Creating a new document with right-click on a template is not a good idea because the document does not remain associated with the template, as can be seen in
Properties. When working with templates, create new documents from inside Writer with
Template to benefit from automatic style updates when the template changes.