Randoms are a problem of their own, so it's not usually a good idea to factor them in. It's no better to say 'just bring friends' since that's sort of dodging the issue, but so is including randoms all the same. Besides, it's not so bad if you're polite-- even less so if you're just plain quiet. On that note, those points sort of cancel themselves out so...
As a rebuttal to your original point, you'll always face a stacked comp somewhere along the line. We can't prevent this in any way, shape or form without removing the random aspect of ARAM, which ruins the whole point.
No matter how we change ARAM, we're really not fixing any particular problem; we're more satisfying a human want for control over the circumstances. Given the number of players queued for ARAM at any given time, I'd say implementing a smurfproofing system would probably have a very nominal effect on the numbers, though I'll only speculate that without actual stats.
I would much rather have a reroll every second or third game than a handful of 'good' champs. All the selection in the world doesn't mean anything in ARAM if you're accumulating a roll every tenth game or so. Nothing will save an all AP team from demise against a team with remote balance.
I think there's enough champs that are solid contenders for ARAM to make this a non-issue. The likelihood of being comp screwed is unlikely to change, just the mindset that you can't smurf-- but ARAM is just as much a matter of being able to use the champs in the right ways, too. There's more to the game than poking.
I think this is probably a problem that seems more dramatic in its moments. Realistically speaking, there are so few champs that are honestly 'bad' at ARAM and even then, virtually anyone is a viable choice in the right hands.
As a counter-stance, and to bring a little more perspective to this issue, do you possibly have a list of champions you'd consider not to be favorable ARAM picks, going by what criteria you built your initial list of choice picks on?
The way I see it, the core of the issue is that smurfs can obviously limit their choices enough by avoiding 'poor' contenders, but there are a lot of excellent choices for ARAM; Ezreal and Morgana are good, for sure, but so are Leona, Xin Zhao, Lee Sin, Galio, Garen, Malphite, Olaf, Karthus, Gangplank, Miss Fortune... to say nothing of champions with strong synergies (Braum/Yasuo, Fidd/J4, Galio/Ori) turning normally viable contenders into unstoppable engines of mass destruction. I would be surprised if more than 30 champs could rank so poorly that they would honestly be a discredit to the team objectively; as long as they're not that, the very extreme cases are the truly irreparable ones, and those will never get removed. That aside, a player's individual skill can go a long way in making (or breaking) virtually any champion.
EDIT: If we did have to select a solution, mine would be to reward non-smurf play-- extra rerolls would pretty much totally leaven the gap, realistically, no matter how it was handled. As little as one extra would make worlds of difference.
