One change I've thought about is just to tighten the definition of what constitutes an advantage in attacking areas, and to extend the time that it takes for the advantage to be happening . At the moment if you are fouled but stumble and get a shot off then your deemed to have received an advantage, even if the shot is nowhere near the goal. Or you pass to another player and then they get dispossessed. If that was changed so you would only be considered to have gained the advantage if you score, win a corner, keep possession for another 15 seconds or so, or have a second phase shot then attacking players might not go down so easily and the game might flow a bit more. Because as it is, there's no incentive for players to try to stay on their feet and keep playing in those situations. Basically it's more like the rugby or league interpretation of advantage.
It might not work at all of course. It might have all sorts of stupid unintended consequences but it could be trialled in friendlies first.