Well, I would say I think sitting out just one play is probably not sufficient in terms of making sure the player is healthy. If he's injured enough to mandate stopping everything, one would think it is serious enough to warrant a more serious amount of time on the sideline.
I'll notice even you said "or two" for a specific thing which is really a non-injury. I would add though you can't even communicate with someone who has the wind knocked out of them immediately, so I'm not talking about just recovery.
A: Player has to recover.
B: Player has to be evaluated.
C: Player has to be determined to be ready for play.
I don't think you can generally speaking, obviously there are some exceptions, be expected to do all of that within the time frame of a single play, unless of course the player is faking. Making it three plays for example on a mandatory basis would both discourage faking injuries and make sure actual injuries had time to be evaluated.
Not really most of these "injuries" happen to guys who really arent really doing anything. Auburn has literally called people out from the sideline to take a fall all of the sudden and it not be a player you have ever seen before. Teams are always going to have a preplanned guy going down no matter how many downs you have.
The only way(s) to stop it is either dial back the offensive rules that make the game fully slanted to the offense, or get rid of the "clock stops on first down" and add "take a timeout away for injury under 2 minutes".