I am fully prepared to get flamed.
Here's the problem from my perspective.
Injuries happen often in football.
When an injury happens to a defender against our offense, and we're "going fast"...conceivably what is the defense to do? They have around 5 seconds after the tackle is made to figure out what to do.
You have injuries where:
1. Players can't move (leg/back/neck/etc)
2. Players can move (arm/shoulder/etc)
3. Players may seem okay, but isn't (concussion), and/or adrenaline carries them for a bit, then they realize they're hurt and fall down.
#1 is obvious, and is pretty easy to tell if it's authentic or not. Play is stopped for real injuries, it isn't an unusual situation.
For #2 they just have to get off the field. Easy, right? What if you're in severe pain? Can the defensive staff find a suitable substitute for you on that short of notice, even if you can make it off the field? Not likely. I think this is why players "fall down", because then at least it forces time to handle these things.
For #3, that's tough. Time is short, so how do you make the right decision? Player health is at stake in situations where there's a real injury. As a coach, would you want to have to make a "get off the field" decision for someone who might be concussed in that short an amount of time?
We go fast. I honestly do not think we are trying to take advantage of an injury situation by going fast. I think we are leveraging knowing our plays that well, and our efficiency to take advantage of defenses being out of alignment. BUT, one of the things that happens is that we are going fast when injuries occur.
As many on here have said, the best solution is mandatory sit-outs for the injured player. Similar to making players sit out for a play if their helmet comes off. For situations where an injured player is actively leaving the field, the ref should stand over the ball to give a decent amount of time. But the injured player still has to sit out, so there isn't incentive to abuse the rule.