ARAM is the perfect place to try new champs. You can actually have fun even if you suck without people raging. In other modes, people have this bizarre belief that the outcome matters (Btw, it doesn't. It's a game. Grow up people.) In ARAM, you can (usually) get a decent group of relatively mature teammates who are actually there to have fun, not get pissy the moment a matchup with completely random players doesn't end perfectly for their team and terribly for the other.
If you think you should only get to play champs you've bought (under some weird assumption that the set of champs you've bought is equivalent to the ones you're good at, which isn't even true for 99% of players) then you've completely missed the meaning of All RANDOM, All Mid. It's about the surprise, not the "do what you do in every single other game all the time." In fact, I'm more interested in the exact opposite idea some people have proposed: make all champions available in ARAM, not just the ones you've bought and f2p.
What you're asking for is basically another game mode. It isn't ARAM and you might as well be asking Rito to remove ARAM.