I was gonna say something snarky about your desire to do FATEs without having to move, but I actually think I get what you're saying finally. You can find this camaraderie at seasonal event FATEs where players have to stick around in one spot for an interminable length of time in order to farm whatever drop there is for turn ins. I've actually had a lot of fun at these (when they're new) since you've got a group of people just kind of derping around. We're trying to make the best of a bad... well, boring... situation.
I think what it boils down to more than just a desire for "exp parties" is for this game to slow down. A LOT. Even making a party to go dungeon with friends isn't quite the same cause you all still need to be actively doing something until you're finished. You can grind stuff, like FATEs, but like you said you're usually always on the move to the next FATE. Amusingly enough, when lulls happen while FATE grinding, people can't stand it and usually spend the time complaining that there's no FATEs or planning to do something else once they're finished in that zone.
That all being said... I'm not sure you can add something like this to XIV without there being a massive torch-and-pitchfork style riot. People are already complaining about not really having the time to do a full run of CM or Prae (for example) due to real life obligations that keep their play time short. It works for seasonal events because they are really optional (or really, you only have to do one FATE in order to actually complete the quest. Sitting around farming them is just if you want all the goodies. I was having lots of fun just hanging around as a Spriggan that I got over 40 eggs. >.>; )
After hanging around that one time, however, I had ZERO desire to do it again for my alts. :/ (I took one other through and just did one FATE for the achievement.) I'm not sure I'd be keen on having this be a regular thing in the game that we actually need for progress. ^^;