I get your point so perhaps the first player of the other faction (or one of the classes, any other idea?) gets also a Button like “Team is ready”.
If booth Teams (Leader, Host, Class, call it how you like ^^) pressed ready the “final” countdown starts. So booth teams can take their time.
To avoid that one team delays the game too long increase the initial timer (2, 3, 4,… Minutes). Perhaps make it depend on in- (less time) or outdoor (more time because of potential more players).
Additionaly as one Team press ready the counter could get down to a specific time (30 seconds, 1 min,…). But I think if I understand you right you would prefer without this addition to keep it fair.
The reason why asking for something like that is because of the nature of technology, different smartphones and the problems that come with that.
Players struggle to connect in time, troubles finding game, connections got lost, app crashes, batteries got empty, you name it.
In a perfect world 20 people connect in no time and the game starts without any problem. In our world we needed 3 tries to start a 1 vs 1 because the timer run out of time and one player was alone in the game.
You don’t have to reinvent the wheel. Just take a look at nearly any multiplayer game for PC and how it is handled there.
My suggestion only counts for the quick game mode. At any other mode there can be circumstances which could force other time countdown “rules” (just to say it and avoid comments like “but at a big event or in game mode xyz” or equal). That would be a different topic