No, bans will not fix this. This will just have ARAM accounts banning champs that counter them. There are also a lot more champs than 10 bans that have overwhelming poke (>50 for sure)
Except that ARAM accounts barely have over 10 overpowered champions, so the proportion of their roster that you negate with Bans highens more the chances of them having a more balanced champion rather than giving all the champions to everyone, while they can trade, reroll twice and use the reroll pool.
Also, ARAM bans are much more versatile in the way they can enhance the game experience: not everyone hate the same champions. You mentionned Jayce, but ever since him getting nerfed, he is much much less of a threat than before. The thing is, if you still absolutely hate him, you can ban him, or you can ban a much more unbalanced champion, or you can ban another champ that you ahte to play agaisnt but who isn't necessarily over the top broken.
Unlocking every champs for ARAM will only fix the ARAM account problems, but will keep the gamemode as the coin flip it became because the overpowered champions are still availlable for everyone and (like I mentionned) there's just way too many ways to get to them even if you initially got a balanced champion. ARAM bans, however, gives you a little bit of control over how fucked you are if the gamble you take fails, while making the game a bit more fun.
Also, there's no such things as "banning champs who counter them" in ARAM bans, this isn't Summoner's Rift, fam. You don't even know WHAT you'll get after the bans, even as an ARAM account, you don't even know WHAT counters them (counters in ARAM are waaay different) and, most of all, the way you get "countered" in ARAM is almost in every cases related to the whole enemy team's composition, it isn't related to a single inidividual (who has high chances to not even make it into the game) that you might ban. So yeah, ARAM bans giving an advantage to ARAM accounts with this argument? Bullsh*t.