Didn't see Pepisolo post, i was writing too long, anyway, here's some thoughts...
Ok, lets see if i can offer an idea and not get too carried away, but more feet on the ground kind of thing.
we had 4 to 6 player co-op where the players go into a lobby and get a 'single' bot, what they get is one of the base exos, not the epics. they can trick it out with perhaps 2 accessories (there are usually 4 in a team of 4, so that is doubling for mp, but i'll explain why that is ok in a minute)
So, when they are done tricking it out they hit a 'ready' button, that just tells the others they are ready to go. First one ready will be assigned the unofficial role of commander, meaning if the other 3, or 4 or 5 players aren't ready, he can signal them with a red 'Hey!' button that simply flashes and honks and lets the other players know you are waiting on one or more of them.
When in the mission, they each control their own bot and go in rotation in the same order as they 'readied in'.
So we have player 1, player 2, player 3, player 4, player 5, and player 6 and then the bots, which can move together or separately, you just put in 4 times as many bots as before.
Am i trying to be stealthy and careful and bob rushes in willy nilly guns blazing and ruin everything!? WELL, we are going to have a talk about that!
So, we need to have some form of 'communication' inside the game, either voice, or chat window and maybe a ping on the map thing. But anyway, i think this could work, its just a matter of the team being civilized inside their game and talking and deciding what they want to do, perhaps in the lobby before they get into the game, or sit in the game and talk strategy and then start moving about.
Anyway, the sequence of players taking turns will work, if some 'single' person (first one ready) has the ability to bump the others if they aren't quick enough. For Friends this shouldn't be too rough as they know each other, but would solve people holding up your game and you not knowing them very well. Also, you could have a default 'Fail' timer, say 5 minutes, that is ticking down and if everyone isn't moving after 5 minutes on their turn it dumps them automatically, and if the entire team isn't ready to go within say the same 5 minutes in setup, well then the team is warned at 3 minutes, warned again at 4 minutes and game cancelled after 5 minutes.
If this is all played on someone's home machine and they are all dialing into one players pc, then all this stuff about 5 minute drop time can be disregarded and the person hosting can call the shots on how long to wait and what is reasonable.
Still with alternating turns for the players, each with a single exo, instead of a whole team, so the team consists of a player having a single exo each, and then pumping up the bad bot numbers to deal with 4 or 5, or 6 people running around a mission.
Also if the missions are randomly built, then perhaps 're-sizing' them for 2 or more people would be a good thing, Gives more room to run around. Of course this involves other things like cover and other things too, but if this goes in later you can take your time and get it to work without the pressure to get this in quick and have it work while players expectations are panting.
Anyway, just my two cents, I think it could work, and if done in the background and slow, it doesn't have to be a pressure cooker, and also, the players in the team have to take some responsibility on how they deal with each other in a game, it isn't all on the dev to figure that out, or make that work. Just give them their turns and tell them to be nice and then throw them to the wolves... er, bots i mean.