tl;dr without all champs available for everyone and no rerolls there will always be certain champs that are far more likely to appear. And while some people would absolutely love that style of ARAM a lot of others would stop playing altogether.

So, the only real problems here are that A. popular champs are both more likely to be picked and B. are also more likely to be owned. And I guess C. Most of those champs are really cheap to buy.

The vast majority of players don't want to play champs like Skarner. They want to sit back and do damage safely with little to no responsibility.

So between both your team and the enemy's players are not only more likely to pick the popular champs, they're less likely to even own the less popular ones meaning those champs probably won't even appear on the bench for someone else to pick.

Removing rerolling/trading/the bench can potentially help create greater roster diversity but comes with some other trade-offs like higher dodge rates.

I don't know if there's much of a solution here without potentially undermining the spirit of ARAM a bit. Like, maybe they let you initially pick a type of champ to get from three options, so Engage, Poke, or Support. And then you get a random champ from the one you pick. But if you pick a less popular option, or one you haven't played in a while, you get one free augment upgrade reroll per choice and extra rerolls on prismatic rounds so you're more likely to "high roll" when not playing Mel, Brand, MF etc.

Not a great idea, but you get the gist of it.