How to balance many champions all at once

EDIT: This would be much more easy to implement after they've given everyone 10 bans.

To illustrate my point, there some cases where the champion is interchangeable insomuch that they're almost never banned* (because banning one still leaves the others as an easy substitute) include groups such as: Malphite/Amumu/Sejuani, Jax/Xin Zhao/Yi/Vi, Thresh/Leona/Nautilus, Nunu/Azir/Nidalee. Okay, that last one was a joke because the similarity is that they're too bad to pick. ######Of course, item changes sometimes give one of the group a distinct advantage, as Cinderhulk did to Sejuani and Guinsoo's Rageblade did to Xin Zhao. And some undeserved nerfs due to item changes may have some champions weaker than subistitutes.

However, there are other champions who have specific strengths which make it impossible for an interchangeable choice. And that is what puts certain champions at the top of the ban list continuously, often times even despite nerfs. Some include:

• Yasuo -- While there are other hyper-carry options, there are none who reach their power spike as soon, are as mobile, have CC AND have a long-duration projectile negating ability. The only other option with even a similar ability to his W has to build tank to make it work, and can't be played viably anywhere but the income-reduced Support role.
• Lee Sin -- He has almost cost-less abilities, mobility to both engage and run, strong enough clear to take the raptors, high base damages, CC, and bonus damage to backliners for attacking the front-liner.
• Darius -- While the other champions in his sub-class have gotten nerfs that hinder their use and ban rate in all levels of play, Darius's incredible early strength which has received no nerfs makes it easier for his players to overcome the few weaknesses inherent of the sublcass. Even the champion that was always supposed to be his mirror, Garen, can't hold a candle to his updated passive.
• Vladimir -- Vladimir has the distinct advantage of being able to buy Spirit Visage for AP. This allows him to have more of his damaging stat, more defense, and more of his "resource" where other champions would otherwise be sacrificing both AP and Mana in exchange for the defense of Visage. And there is only one other option for the costless in-lane sustain to match Vladimir's: Aatrox. Of course, Aatrox is melee and has a slew of other design flaws that make him either the best or worst option. For the last few years, he's been the latter.

And then there are champions who have strengths that are specifically abused in professional or otherwise high-level organized competitive play:

• Ryze, whose ultimate is a game-changer with no equal. Tahm Kench gets closest to his ability, but can only bring himself and one ally where Ryze can bring the entire team.
• Shen, another case of a game-changing ultimate. With Pantheon having his global range long since removed, Shen has become the only option for pro players that want that strength from a top-laner. However, Twisted Fate and now Galio have become problems due to their ability to reach either lane from mid.
• Rek'sai, who can provide free vision to players with the concentration to keep tabs on all her holes. ( ͡° ͜ʖ ͡°)

Mirror lanes should be allowed in draft. If it were made possible for both teams to have any one of these champions (and any that I didn't list), then we could see a state of balance that hasn't been seen in the history of this game. It may take a while to get used to, but eventually people will see that they don't have to ban [champion X] if they trust their ally to be better at that champion.

Important note: The change I'm asking for is solely to remove the lock-out of a champion that the enemy team has picked. There would still be bans and it would still be a draft pick.