Quite a good idea. I don't know whether everyone would welcome "overworld" chatter as part of the base game, but it would definitely be a welcome mod.
I am also using Histidine's Combat Chatter mod, and it provides some useful info at times, regarding ships running out of missiles, or getting overloaded, which sometimes help in giving commands to save said ships just in time, or change the formation a bit, so that they get better cover. Some similar responses or comments would be nice from, say, the nav officer, saying, that "Fleet X is pursuing us", or "Pirates are maintaining following distance", or something along those lines. Also terrain could affect the frequency of chatter. So while just travelling, you could get idle chatter between ships or just the crew of your flagship, but entering an asteroid field could prompt related orders, preparing the crew to brace for impact. Going dark could result in radio silence, etc.
All in all, sounds like good stuff. Space shanties would also be awesome, though implementing them is another matter entirely. Reading chatter is one thing, reading a song however is not necessarily much fun. The only idea that comes to mind (short of replacing the game's music) is to make a few second recordings of songs, then play them when certain chatter messages come up, the same way a sound effect is played when an expedition comes to one of your colonies, or you complete a mission.