There's a reason this idea is shut down every time it comes up (it was pretty much a weekly topic in the old ARAM section of the forums). And I actually hadn't even thought of your scenario. The two biggest issues are when conflicts occur, or how people could abuse this.
What happens when several players dodge the same champion, and then they wind up paired together? Or to your point, if the free champs change after they dodge, making that champ no longer available to them?
But I don't think the people that propose this think much of the possible abuse, as well. People with the good champs could start dodging if they don't like what's on the rest of their team. So now they're guaranteed that same top-tier champion, but they have a chance at getting an even better comp. Take this to the extreme, and a group of friends could do this until they're guaranteed a team full of top-10 champs. It's worse than running ARAM smurfs...
I'm right with you that I'd love to see all champs opened up for ARAM. It's something we've been begging for for ages now. And heck, I'd prioritize that over doing anything more about dodges. This could be partially due to me having played through the times when it was still a custom game played on SR, where it was common to sit through 10-15 dodges before getting into a game. The 2-3 we have now really doesn't bother me.