As somebody who mains Azir (and has since the end of April) I think I have a good idea as to what Azir's identity should be, what it is right now, and what changes would need to be made to accommodate it.
Azir right now has a lot of things he can do. His Q and W allow him to poke somebody substantially, his E gives him some mobility (which can become very high mobility if used correctly), and his R can completely change the landscape of a fight. His passive is simply an AA steroid (one that is made to work with Azir's itemization) plus the turret active (which is cool af). Right now if played correctly (which fortunately takes a fairly decent amount of time to learn how) he can respond to just about every situation.
Azir intended niche is a mage who specializes in control, sustained magic damage, and safety. I would define safety as strong waveclear, decent (but properly gated) mobility, and a CC that gives you a chance at redemption from a bad situation. The very nature of his kit gives him lane bullying because of his W, but that spell is so integral to Azir that it probably cannot be modified much if at all. Azir should be strong at killing tanks much like an ADC, and through his Ult and placement/movement of his soldiers he can control the pace of a fight and how it plays out if the player is adept at Azir. What Azir shouldn't be doing is taking most of a champion's HP with just a Q -> AA once he has a Deathcap.
The solution to this? Well, Azir has a lot of multipliers throughout his kit. Sustained damage generally scales through these multipliers. For Azir, these multipliers are attack speed, CDR, AP, and magic penetration. Additionally, striking enemy champions with more than one soldier with Q or an auto-attack makes the second strike deal reduced damage (but still acts as a damage multiplier).
A good solution (without being too drastic) could be reducing the AP ratio of his Q further (maybe down to 30%?), lowering the AP Ratio on his W (down to 50%), and then increasing his base attack speed from .625 to something more commonly found on an ADC (.658 is common). If needed, they could also increase the attack speed he receives from his passive. This would encourage item builds, runes, and masteries to include some source of bonus attack speed outside of his passive. You can still easily achieve high levels of AP without a Luden's Echo and the AP item changes on PBE will also be a catalyst for this.
Personally, I never build Luden's on him (unless stupidly fed). I prefer to purchase a Stinger after building Morellonomicon -> T2 Boots -> Deathcap. I then proceed with a normal build until my final item (where I would finish Stinger into Nashor's Tooth). I generally end up with 1.5-ish attack speed, 720 AP or more, and 40% CDR (over-capped by 5% because of masteries). I may not be able to poke as hard (keep in mind, your poke is still EXTREMELY potent), but in teamfights I'm feared by tanks and squishies alike. It also just feels more natural, more how I feel Azir should play (and thus became my preference).
Also, they could remove the tower-damaging active from Azir's W if they wanted. It feels like unnecessary power in my opinion, and doesn't really contribute too much to Azir's theme.