Why are we even playing together as a group?
Because we're friends who enjoy RPGing together? That's always been my main reason.
I've run games where the PCs had different goals and concerns, but I've never even heard of a game where there was no "adventure" with all the PCs participating in it.
As I already mentioned, I think Apocalypse World is the poster child for this.
The last time I ran Cthulhu Dark, I told the players I wanted us to play in late-Victorian England. Character creation consists in choosing a name and occupation: one player chose an American journalist visiting England, reporting on imperialism for a left-wing paper; the other chose a butler sent to London on an errand because his master was indisposed. I started with the journalist and introduced a mystery/lead; I then cut to the butler and introduced a different lead; then back to the journalist, where I had a fire start in the apartments he was visiting; and then to the butler, who - as it turned out - was next door to the fire. It didn't take long to intertwine the mystery of imperial dealings in Bohemia and East Africa and the mystery of the indisposed master: the point of intersection was
were-hyenas. The two PCs crossed paths more than once, but never actually worked together.
In my Classic Traveller game, which as I said is a weak case of the "no team" phenomenon, the players control positions each with multiple characters: at least two in each position are unequivocally PCs, and then the rest in the position are semi-PCs bleeding into NPCs. One of the player's main PC owns the starship, and has his goals, which tend to involve raising revenue but also obtaining technology for his homeworld. A second player's main PC wants to master psionics. A third player's main PC wants to learn about aliens.
As referee, I try and create situations where these various goals are in play, and potentially at cross purposes. In our more recent sessions (not super-recent given the pandemic and its lingering consequences) the various characters have been spread across different locations - on various worlds, in various vessels, etc - and the action has involved cutting between them. The starship-owning PC's position includes a NPC from whom he won the ship in a bet, and who is his lady-friend; she is also a surgeon and bio-weapons experimenter who is breeding Aliens (TM) in the ship's sick bay, which is a source of concern. The psionic-aspiring PC has a habit of law-breaking and causing trouble; most recently she blew up a noble and retinue from another nearby world, which seems apt to cause blowback. Some of the PCs have also had recent word that an Imperial armada, which has some knowledge of their toying with psionics, is crossing the galactic rift in pursuit of them.
I would describe this game as involving a series of unfolding and ramifying situations, cutting across the different player positions in various ways (eg one of the experimental Aliens badly mauled another PC's NPC girlfriend; she is now in the sick-bay being "tended" by the NPC doctor). Various characters work together from time-to-time, as suits the players and the situation at hand. But it's not much like D&D-style party play: there's no common "thing" that they're all committed to achieving at any given time. It started out a bit more like that, but as the player positions built up (due to the PCs recruiting various people as they went along) it changed into what it is now.
