If I understand correctly, Users are complaining about losing unsaved data in the query tool and not about data output or session state. Hence just reopening the query tool with only data should be suffice.
I'm sure that will suffice for 95%+ of users. The ones I'm concerned about are those who (for example) have done SET search_path = ... and then performed some destructive operation that worked as expected because of the earlier SET, but might cause data loss or unexpected consequences if run without the SET.
Granted, that class of issues is likely to affect only a small number of users in reality, but the consequences could easily be data loss.
Just a thought from an onlooker. My main request for this feature would be to re-open query tool tabs with the contents restored, whether I had saved the files or not. I don't think I would expect to see any results populated in the results grid, nor would I expect temporary SET operations to also be restored.
I am curious about a comment about saving data on the pgAdmin server. Would that mean sending the contents of my files to a server outside my org? Forgive me if I am misunderstanding.