I wonder if it's some kind of design choice or even a technical limitation that made saving only possible in the player's room.
That aside, there is a long-standing bug with saving where reloading a save can "fast forward" certain events. An example of this would be saving on the first Friday, going to church, seeing the dialogue between just the family members, quitting the game (without saving), reloading that same save on the first Friday, going back to church, and then instead of seeing just the dialogue between the family members you now have Alice's dad show up as well as Alice herself at the end. In other words, you've just seen the second Friday's event on the first Friday.
It can potentially screw up the pacing of events so in my opinion until that bug gets fixed I think the devs should hold off on allowing us to save anywhere for now.