For teamgames, a system where some of the players in each team can get a random civ should work, if that would be good enough for players like you (eg if in a 3v3 2 players of one team and 1 player of the other team take the option, 1 player of each team gets a random civ).
For 1v1s I’d like to see a map in the pool with “force random civ choice” associated with the map. People who don’t want to play with random civ can ban the map.