Holy moly, I wasn't expecting such a long reply.
Long story short - You don't balance to the average player.
The average player isn't what breaks a character. The average player will play a character and do average with the character. There is no need to balance what is already balanced.
Let's use your example - Azir - You are seeing him in Bronze I-II. Now, the Azirs you have played with have 'done terribly'. Is that the fault of Azir? Is that the fault of Azir's complexity? Is that the fault of the player playing the champion?
What would 'lowering Azir's skill floor' accomplish for players in Bronze I-II? Players will still use their abilities incorrectly, they will still make basic mechanical mistakes, and they will still do the wrong thing, but that isn't because Azir is overly complex, that is because Azir is being used by a Bronze I-II player.
Let's use a character with a very low skill floor - Garen. You give Garen to Zionspartan, and Zionspartan can make Garen look like a god. A slow, clunky, lumbering god, but a god nonetheless. Now, give Garen to a Bronze I-II player who has never played Garen before. Suddenly, Garen looks confused and like an ineffective champion. Garen literally has no skill floor - He is as basic as you get, so why is the Bronze I-II player not excelling as Zion would?
This is the same fun ideology that caused something like Bronzodia to exist for so long, and for the silly bans / picks it represented to happen - Things like Blitzcrank, Amumu, Malphite, Shen and whatnot were Universally banned in Bronze because they have theoretical 'Low skill floors' in which anyone can simply pick them and do moderately well with them.
However, everyone forgets the reason why this is the case - It is not because any of these champions have low skill floors, it is because these champions are being played against inexperienced players who do not know how to avoid what they do.
You could increase Malphite's skill ceiling by adding more counterplay, but why? What he does will still be effective against weak players, and less effective against strong players, so you are rebalancing Malphite for something entirely out of anyone's control - Poor players cannot process things properly, good players can.
This all leads back to the players - Low elo players will make many more mistakes, regardless of champion picks. The skill floor or ceiling of a certain champion is redundant based on the fact a player who does not play very well will not play any champion very well, whether that champion is Lee Sin or Azir or Malphite.
Comparing it to team fortress is comparing Apples to Nuclear Fission Techniques, TF is balancing classes, League is balancing over 100 different unique champions.
Basically what I'm going for is this:
Champion play at low level will always be limited by the knowledge of the player, which will always be limited itself by lack of play, game mechanics and practice
Balance will always be tailored to the higher end of the spectrum because that is where balance problems tend to arise on the first place. Both exposure and visibility contribute to these problems where recognizable figures point out the problem and exactly what causes it.
That Bronze I-II player who is a terrible Azir isn't suddenly going to become an incredible midlaner if you give him Ahri, he's still going to be a Bronze I-II player who is playing Ahri. He's still going to miss skillshots, he's still not going to ward, he's still going to spirit rush into teamfights 1v5, and he's still going to chat in all that the enemies are lucker dogs.