This.... is ridiculous. Azir was designed as the AP equivalent of an ADC in that he's an auto-attack based, late game carry who happens to build ability power instead of attack damage. His soldiers are invincible because they're the means of delivering his magic damage based auto-attacks to his enemies. Making them killable would be like countering an ADC's attack by right-clicking the projectile to knock it out of the air. Yes that would look cool, but it would effectively destroy the champion's ability to do their job properly.
I've seen the Heimer-Turret comparison in this thread, but there is actually a key difference between them. Heimer's turrets have their own health bars because they remain indefinitely and attack independently of Heimerdinger, leaving him free to cast spells, auto-attack, use items, etc. They're more comparable to Azir's passive turret, which does have it's own health bar and a set time limit for when it can be up. Azir's soldiers on the other hand are auto-attacks, they can only attack when Azir orders them to, can only attack the target he's clicking on, can only do so when both the target and Azir are a certain distance from them, and can only be out for 9 seconds.
You mention "This change would give some counterplay options for his opponents, defined strengths and weaknesses as well as having windows of power" but the thing is, Azir has all of those currently (it's why they launched with invincible sand soldiers and never changed it after two years).
Counterplay: stay outside of the range of his soldiers, he only has one ability to move them, so punish him for missing you with it. He's a champion you beat through superior positioning. Azir also builds relatively little mana, and can run dry early on if he frequently combos, punish him for it. If he has multiple soldiers down, then he has no escape, move around the soldiers and dive him. Always coordinate with your jungler (this is good advice in general).
Strengths: Powerful siege and defense potential, strong burst and DPS come late game, potent engage/disengage with his ult.
Weaknesses: Smaller mana pool early on, 3 WQ's is enough to effectively drain it. A weak early game in general, besides his Q and R (and technically passive), Azir has no damaging abilities and is reliant on auto-attacks. The result is that he can be effectively bullied early on if you play well. This is also why Rylai's is a core item on him. Since he can only move his soldiers with his Q (and not independently of each other) he's only able to get off one, maybe two, autos before his opponent simply walks out of range. Alternatively, play a bruiser or a tank and completely destroy him once you buy a
(this tactic is only recommended if you know the people you're playing with well). He's squishy, and unlike actual ADC's has no way of regenerating health through lifesteal or spellvamp. As a result, divers and assassins are quite good at dealing with him later on. While he can disengage with his ult, many dashes (such as Nocturne's Ult) cannot be stopped, which allows them to kill him. Azir does have his own dash, but using it requires him to disengage his soldiers from the fight as well, effectively halting his damage. Finally, simply don't fight him in his soldiers (or near a tower) and he won't be able to deal with you.
Windows of Power: One of the most overt examples in the entire game: his number of soldiers. Azir slowly acquires soldiers (time reduced with CDR) and can only hold two at a time, though he can have any number out on the field at once. The most soldiers he'll have down early on is two, scaling up to 4 (and at a stretch for half a second 5) come late game. The more soldiers he has down at once, the stronger his potential offence is, but the weaker his potential defence becomes as he won't be able to dash away without a soldier in reserve. Azir is at his most dangerous near a turret with 4 soldiers down, and at his weakest alone with no soldiers out.