There is very little reason to proceed into the match if you don't have a) the champion you want to play or b) several of the S-tiers on your team, because you can expect 2-3 of them on the enemy team.
Because of alt-accounts, re-rolling, and trading, it's remarkably easy to get a busted Champion or your main. The vast majority of ARAM matches, in my experience, are over at the loading screen. So as mind's well shut that down before it even gets there if you're not happy with what you have. It's not about winning every game, it's about not playing matches you can't win, which is going to be the situation in 80-90% of ARAM matches.
I don't particularly like it. But "Random" in this game mode means some assortment of Xerath, Brand, Lux, Vel'Koz, and Ziggs, Veiger, etc. are in every single game. And if you don't have some of them on yours, you can bet they're on the enemy team, which means there's very little reason to play that match.