I think Competitive ARAM (i.e. Ranked) would work better than ADAM. There simply aren't enough changes to regularly shake up the Draft meta. It'd just devolve into "Pick/Ban these 20-30 champs every game" indefinitely. Additionally, there's significant overlap with Summoner's Rift in terms of skills tested.
In contrast, the randomness in ARAM promotes a greater variety of champs. Furthermore, the randomness also tests generalist skills, something that none of the existing ranked options test. Rather than encouraging specialization (e.g. Mid main or Riven OTP), ARAM encourages developing a wide set of skills. By requiring a different skillset, ranks earned are more meaningful.
As for addressing the "It'S jUsT rAnDoM cOiNfLiPs" argument, there's several ways to address that. The recent balance changes have apparently already started narrowing the disparity between relative champion strength. Bans would further aid in removing any (perceived) outliers. Going even further, ARAM could draw from card games, another competitive random game, and use a Random Draft in champ select. As with the existing Draft queues, players would take turns picking their champ. However, rather than picking from the entire champ pool, each player would be limited to a handful (say, 3) of random champs. This change would add yet another skill test by including drafting while still respecting the spirit (and benefits) of ARAM.