First round: brown gets a good set of connected areas. Second round: "flag brown". Brown gets an additional advantage in that he knows he's at least not 7th and an indication to other players that he is likely the winner. Other players follow in flagging as they're threatened by brown. This snowballs into a 1st for this player. The game is decided in the first round. Not good.

There's gotta be a good solution to this.

There was a similar behaviour years ago but it had a critical difference. In the second round someone would say "truce brown?" or more subtly "brown, we cool?". If accepted the two would hold off until as a team they are securing the victory. The critical difference is that the snowball of people trucing brown didn't happen. Instead other truces would form and you would have a sort of team game. Much better result.

Is the first version because of the flagging system?

How do you get that third person to not also flag brown but to form an alternative outcome to the game?