You have to balance treating legitimate injuries and preventing faking. Holding a player out penalizes the player; this rule tries to penalize the team.
The thought is that a player who is really hurt will stay down or go down quickly based on the previous play. It’s hard for a defense to get in a “fake the injury” call before the ball is set. This rule encourages player safety, too. If someone thinks they’re hurt, they have to stay down—if you try to fight through it, you may cost your team a timeout if you go down later.
The rule is targeted at the worst flops: the Kiffin flop, the Bama nonsense from last year’s game, etc. Penalize the team for a coach’s call. Sure, a player or team might still fake injuries to buy time, but they have to do it immediately and the worst fakes will be eliminated.
Keeping a player out for a series puts too much on a player. Imagine Mike Matthews getting his wind knocked out on an end-of-game drive with no time outs remaining and needing a TD to score? Do you think he has to come off for the whole series? It’s a real injury but a quick recovery. Or imagine a QB whose head hits the ground in the same situation. He has to be evaluated for a concussion, even if he plainly does not have one—evaluation takes less than a minute in most cases. Should the drive depend on the backup now?