Well, here's how you do it.
ARAM already gives extra points towards rerolls based on your champion pool, which means players who own more champions get to reroll more often than players with less champs.
So let's say one player has 36 champions in their pool, and another has 100. The 36 player gets 36 points towards rerolls while the 100 player gets 100. But if the 100 player was able to pick 36 champions they could draw from in ARAM, then they would get 36 reroll points instead of 100.
This helps the ARAM players, both with high and low champion pools, because it means you're less likely to get an all-melee team and lose by virtue of which champions are picked. It also means that players with high champion pools get to play more ARAM without getting flamed every game for bad champion draws (which does happen, ARAM players flame just as often as other players even though it's not a serious mode).
This helps Riot, because players are more likely to buy champions (with RP) with no fear of drawing them in ARAM. This means more $$ for Riot.
And trust me, you don't want every player to have every champion in their champion pool for ARAM. Imagine getting a noob who sucks at League anyway getting a high skill champion that they don't own and thus have never played before.
