I think the main problem is that diversity in that game mode would suffer alot.
Champions are not really balanced around ARAM, meaning that there are many champions which are really strong in that game mode (like marksman and artillery mages, basically everything with good range), while other champions suffer alot there (for example juggernauts).
Everyone would pick the same OP champions for this game mode and you would see the same teams all the time.
Sure, Summoners Rift you see the same champs many times too. But at least Riot tries to balance all champions for this game mode. You see shifts there. But in ARAM, it is basically determined by the game mode itself which champions are working and which are not....