The basic concept of Ranked ARAM is super appealing, though the specific details are tricky.
Unlocking all champs for the ranked queue would be a nice step to avoid curated accounts abusing champ distribution. Next, having a higher account requirement for ranked (e.g. summoner level 45, 40 champs owned) would help avoid alts and people taking advantage of free champs. Additionally, I'd personally prefer if champ select used a new random draft format where players are shown a few champions and have to pick one to play. It'd make champ select feel less passive and create another skill-test while allowing players more control. Champ select time wouldn't be unreasonable either, since pacing and flow would effectively be the same as the draft format used by the other ranked queues. Bans, reroll availability, promos, and so on could also be adjusted as necessary.
Unexpectedly, the single biggest issue with Ranked ARAM seems to have nothing to do with the mode, but rather players' understanding of randomness and ranked. As shown in this very topic, there's a strange misconception that RNG and randomness somehow invalidate the competitive standard of ranked. If that was true, card games (e.g. poker, M:tG) wouldn't work. These game function by having a large number of games balance out the variance from the randomness. Consistent skills and effort are more impactful than random incidents in the long-term. Coincidentally, ranked in League does the exact same thing. That's why "Oh, I can't climb because my team's always bad" is always treated as a laughable excuse. Moreover, with the ranked splits and position ranks this Season, Riot's been further encouraging playing a large number of ranked games. Arguing that randomness disqualifies competitiveness might be reasonable for tournaments or other events with significant time constaints, but that simply is not the case for ranked.