This is because he requires two items to be effective, Rylai's and Nashors. During then, Azir has no pressure on the game, he is suppose to farm and team fight. Before all the nerfs, if Azir won his lane, got a kill or counter jungle killed the mid-laner/jungler in a skirmish with his own jungler, Azir 80% of the time got the first tower, now he does not.
In most ELOs teams get caught, do not have great team compositions built around Azir, or have short-duration team fights. Azir does not shine in those scenarios.
He is basically a worse version of Ashe with an AoE knockback that requires less gold to do the same job and has no reliable engagement tools.
However, Azir is a Magic Damage Marksman, so tanks cannot easily build around him without giving up armour. Hence, good for a physical damage focus team.
Ever since the nerfs to Azir's tower pushing [E], he became the team fight champion.
He stopped being a great duelist when he lost his Knock Up, and had his Shield Amount Reduced. He cannot compete against Assassins or trade very well.
Furthermore, with the reduction to his soldier’s range, he cannot compete against mages like Viktor, Ziggs, Xerath, Syndra, Orianna, or (Yes) Viegar who can dish high damage from afar.
However, because he is a mage, he has a lot more defense mechanics that allow him to survive compared to an Marksman like Ashe. Zhonya's, E, Ult, Barrier/Cleanse(if taken) and mobility.
In Pro Play, teams are built around Azir, and tower pushing is important hence the removal of Azir's tower damage. Ever since then, Azir has been moved to "team fight mage".
If you compare Azir from the view of a team fight mage, there are a lot more mages that do the job better, Syndra is the best example because she has an AoE Stun, good wave clear and High Single Target Damage. In Pro-Play, you want to instantly delete/remove threats from the game before the enemy can react, if you cannot, then you provide strong team fight CCs. Azir cannot do either and therefore a lower tier mage in that respect.
The only way to make him more viable is to remove his shield and give him more damage. Or, remove his passive and rework it.
Right now, his shield is worthless because a good Azir never engages into the enemy (unless he must), and he is always in the back line. Also, his passive does not contribute to map pressure, and never useful for attacking or defending. Teams have learned to wait for the tower to disappear, or avoid it completely because they know it has a long cool down.
Still, he is not bad, he requires very certain team compositions that allow him to shine. Although, I am not sure if you want a mage that does sustain DPS over a mage who instantly deletes/removes a threat from the game. [That is the job of the ADC, Jungler, or Tank]
A rework/update/buff is required but he aint that bad.