I am presuming @xiota's suggestions below only apply if the document has no existing path and that "previous save-as path" is the path the last time the save-as dialog was used, not the path of the last saved file.
Yes.
Makes sense except should fallback to the prefs "startup path" if there is no previous save-as path then to GTK default behaviour not home.
On my computer, the "startup path" is blank, and the GTK default is probably the home folder. So whenever I refer to "home", substitute with "startup path" and GTK default.