I do think it'd be nice, and quite frankly a little more accurate, if Riot expanded the champion mastery system over to ARAM.
If Riot were however to expand champion mastery to ARAM, they'd need to adjust the way it works. They'd need to grant you less mastery points when you won matches since the games are generally shorter, don't have the complexity of Rift matches (ie objective control, grouping vs splitting, warding, etc), and an overall different dynamic requiring you don't play the champ in the mode they're balanced around and thus don't need to know them as in-depth as you otherwise would.
That being said, I believe that it would be unfair to mains that have hundreds of games with their respective champs if someone played a handful of ARAM games and achieved the same level of mastery with far less skill (going back to the dynamic gameplay part). Because of this I wouldn't be surprised if Riot imposed a cap on the level each champion is able to be given through ARAM, such as no champion being able to go above level 2. You would have an understanding of how the champ's abilities worked, get a sense for their stat growth, and get experience as far as teamfighting in a lane goes, but that still leaves a lot of this game's depth to be experienced and "veteranized" through the specific champs.