Alternative to a separate tab could be to list the templates in the "Select survey to copy:" list under "Templates", with a button to enable "copy as template" so when to press copy the new templates will be added that way.

However a separate tab would likely make it much easier to relate to (copy vs template) and having templates listed separately would be also much cleaner (as one list can grow easily with creation of new surveys).

In the bigger picture we might want to consider whether this would be beneficial for a built option also for the question set templates vs Import questions.

Although I would think this would probably operate differently as this is inside a survey and might only suit on a user level (question sets available of all surveys created by the user or question sets that the user copied specifically as templates).

The overall survey question template is what likely more users would benefit as there are always some frequent used surveys.

Templates probably shouldn't show up in the survey list, thus separate tab is likely most ideal.

Looking at the permission model, i realised even currently there is no work around, because even when using a group it actually only adds the users individually (not as a real group per say) and thus additional users added later on won't be added automatically in the survey permission.

Perhaps best would be a separate template flag to be implemented and an exception therefore in the permission model/checks to allow any user to access those surveys with read / export rights.

lots of great ideas here. I'd rather go to something like "import survey template as set of questions" (pretty similar logic as "import file as new layer" in graphical software )
so creating a survey group of survey templates, and then adding a new item in the import modal.