I see what you mean and I have experienced it myself but I don't think it would really work, here's why I think that:
We would need an admin to be on and make sure that everyone is ready, maybe even 2 admins or more.
We would need to create rules about waiting at the start so everyone can start together wich complicate things because people are gonna have to get used to those rules.
It would be very complicated to keep track of every player loading to make sure everyone loaded so the game could start.
The same way late loaders get bothered that quick loaders get to claim assets the quick loaders would get annoyed that they need to wait a long time.
And lastly and more importantly:
Bots wont wait for players to be ready, we would put ourselves into a disadvantage every game if we all waited 5~10 minutes before every game.
Coop is a place of a more relaxed gameplay so I don't see that problem being in high priority, in fact, that is one of those problems that don't really have a solution, it might have but not as fun.
At this point do we really need briefing? we are fighting on the same maps against the same enemies everyday, there isn't much planning to do against such a predictable enemy compared to human players.
Also I don't see the connection of "reduce base rape" there, base rape is against the rules in either way, having a briefing or not. As long as there is an admin online and helpful players reporting those who break that rule we can fight that problem.