I have a pretty sizeable group of friends who like to hang out and play RPGs together. We've done a handful of games, ranging from crunchy stuff through rules-lite games. But one common theme is that our games can go pretty slow. We're a group of 7 most of the time, which means 6 players + 1 GM, which is can make stuff very slow. Iterations of D&D all suffer from this, and it gets really bad in games like Dark Heresey for us. But even tropey, free-form games like Blades in the Dark and most stuff on the Apocalypse engine feel sluggish, since they're so tightly based on the Player-GM feedback cycle.
The only types of games that I've found play quickly with a large group are the light, beer & pretzels games like Everyone Is John, Paranoia, but those don't extend well beyond 1 session. I also like the higher-stakes, longer-form RPGs where the players can shape the world at scale.
Are there games that can support both longer-term campaigns AND large groups, or am I looking for the impossible here?
With larger groups I tend to stick to less mechanically complex games.
Most OSR games can be run on the fly with any number of players. I had a fixed group of 9 run through Keep on the Borderlands, with 1 or two extras jumping in for a session here or there.
My absolute favourite is Savage Worlds. It'ss great as the maths isn't tight and "balancing" an encounter is just a matter of throwing in more mooks, throw in a wild card per 2 or 3 players. It can fit to any setting, though I strongly recommend Deadlands.
My close second favourite is Call of Cthulhu, which I've run with 8 players. There's not a combat focus so sessions are unlikely to get bogged down, and even then, most combat actions are a simple contested roll. Investigations tend to resolve as people splitting into pairs and following different leads; two go archiving at the library, two visit a sanitarium patient, two head over to the local paper to see if any stories have been published or even blocked by an editor, two stake out points of interest.