Yes - SCs (and players!) need a common purpose. Most groups don't have that - I only have that in two of my group. In one all of us have to fulfill a pact with a powerful fairy queen (which we inherited) so we know we have to do the tasks inflicted on us or else...
The other group is a more or less open group: A "failed region" on the edge of a realm after years of war with evil beings and necromancers, full of war lords and mercenary sitting in this castle or that walled village. We defeated the necromancer ruling in the former county capital and try to bring peace and order to the region. What's different in this group is that there are permanent players with chars that have jobs in the administration etc. (herald, wizard, squire, guard captain) which have monthly "council meeting" sessions and even play by email (like the herald seeking information about the city guild's remnants). Then there are semi-regular sessions (where the admin staff often have "commoner" SCs) with changing players where you do small quests, comparable to PFS adventures (visit a haunted mill, clean up the cellars of the castle where the necromancer left some suprises or just sit in a tavern and fluff). In this group we all have a common task - establishing peace and convincing the wary population that we're the good guys and that we can achieve that goal. Even with varying players and SCs it makes the game play more profound.
One of my favorite game designers in the video game realm is Spiderweb Software, who have done a great job in their dozen or so indie RPGs of giving players a good compelling reason to follow the plot, usually through violent coercion. It turns out that being oppressed by an evil empire is a good way to convince people to do things.
Interestingly, the group that I've had the best experiences with is a Discord server (this is where the Only War games happened). While I've generally played with the same few core players, the fact that there's this ongoing narrative has really helped to keep people pushing forward, because you can't just go off on your own to do what your group wants to do. Someone else will have something going on that is important. I think this is more or less analogous to what you describe.