"Only pick and ban rates are analyzed when nerfing champions, because that's the easy way to see if players feel that champions are over powered (it's not because players simply enjoy playing those champions over others)." - A support main that has sympathy for Syndra, Fizz, and now Azir players.
No, win rates aren't irrelevant. They are just part of the equation. You've also got pick/ban rates obviously. Then you've got K/D/A, damage dealt, healing, crits dealt, cs, towers, dragons, barons, not to mention what items are most built on a given champ, how well they work with certain other champs, what sort of strategy is most popular with a champion. There's numerous factors to consider when thinking of balance.
You can just watch general gameplay and watch closely over many games to see how a champion performs instead of judging based off of numbers.
Win rate is irrelevant because of pick rate. A champion like Lee Sin gets picked 160,000 times while Karthus only gets picked 3,000 times. Is it truly fair to look at win rate when there is such a disparity between the two champions?
The best example is Urgot. Statistically, he has the lowest lose rate. This is because he has the lowest pick rate as well. Not many people pick Urgot in ranked, therefore the rate in which he loses is significantly lower than someone like Lee Sin.
There are several things that can send up flags for balance. Win rate is one of them. Pick/bans is another. K/D/A is another. General gameplay can as well. I assume the last one is why Azir got flagged.
