My solution: if play has to stop because you're rolling around, you are clearly injured and must be substituted. This would fix a lot of the fakery I imagine.

Again, sounds like someone that has not actually played. There are times where you just need a damn minute. The rule saying that if you require having someone come look at you on the field means you must spend at least one minute on the touchline before being allowed back on is fine. The ref accounts for that time of stopped play to extend play in what is known as stoppage time. There are plenty of times where the player is not actually looking to stop the play, and the ref doesn't stop play. There's other times where you actually need the magic spray. Of course, there are times where your team just needs a break in play to regroup.