ARAM is probably the worst mode for earning chests on, and not just because you don't get to pick who you play. (Although that doesn't help either)
In ARAM, some champs are just plain busted. Those champs tend to have over 20 kills, few deaths, and have the most damage to champs in the average game. Getting an S rank on those champs is exceedingly difficult, because kicking all kinds of ass and carrying like a god is the average. I personally suspect there are a select few champs that if you die more than once the entire match, your chances of getting an S are nearly non-existent.
Also, doing well in ARAM sometimes isn't just about how good you are; but how well both you and your opponent's rolled. If you're a Rammus against five auto attacking AD champs, that's probably going to be at least an A rank game even if you've never played Rammus before. Some team comps feed right into your strengths, like the enemy team not having hard CC to deal with your roll of Yi or Kat.
The reverse is also true: sometimes you get a team full of five squishy champs that do best when peeled for, against five champs who are strong divers and heavy initiators. It's going to be difficult to get an S rank as Koggles if every teamfight is a Malpite ult into a Hecarim ult into an Akali/Irelia tag team diving on your face while the rest of your team runs back screaming "Damn, glad it wasn't me".
And finally, there's no roles in the ranking system in ARAM. On SR, supports get huge allowances for what they do. It doesn't matter if they had a lack of damage or a bad K/D ratio or terrible farm; they can still earn an S rank for great play. In ARAM, even if you roll up a support, you better be ready to farm those creeps and do some damage if you want that S rank. Even Janna is going to have to tornado those minions and grab that deathcap if she wants a box a lot of the time.
Honestly, not being able to choose your champ is probably the least of your worries if you're trying to earn chests in ARAM. Sometimes, RNG just stacks the deck too much against you regardless.