I think that is indeed why 'Lower Decks' worked because we can all relate to being on the bottom rung and wondering what happens when our own bosses go into meetings that we never know the content of in full. It's a very relatable premise.
But then again in Trek we're used to spending time with those that all the action happens to. But it's like the Enterprise problem..............not all the excitement happens on the Enterprise (of any version) at all and other starships must have their own adventures to tell.
You own idea of a freighter series would feature the John and Jane Does of the Federation itself, albiet in a setting we've never seen before. So, I think there are avenues that could be considered, although not all of them would end up good TV shows. I do tend to believe soap opera elements would be unavoidable in the end as Star Trek is itself in my opinion Space Opera anyway and all the shows have soapy elements sooner or later.
But anyway, it would be interesting from an experimental perspective just to see if the 'City on the Edge of Forever' script would play using each subsequent show and each subsequent Captain/Commander just to see if it would have the same effect as the TOS version.
'If the Apocalypse starts, beep me!' - Buffy Summers
'The sky's the limit.....' Jean-Luc Picard, 'All Good Things'
courtesy of Saquist