'Sin bins' for football?

Having watched the recent Manchester City v Arsenal match I wonder if the automatic dismissal rule could be amended.
After just four minutes a City defender, who hadn't previously been cautioned, was sent off for bringing down an Arsenal player. Taking place so early in the game robbed the game of its meaning and deprived the paying customers the spectacle they had every reason to expect. They were defrauded because the game was now heavily biassed towards one of the teams for almost the entire match.
How much better it would have been if the player had been 'sin binned' for ten minutes as they do in rugby matches. Punishment exacted and the spectators not swindled. If two such offences are committed by the same player, then sure enough, send him off for the remainder of the game.