You have your own dash for a reason.
Selected assassins, e.g. Fizz and Zed, has two gap-closers: you used ult to peel one gap-closer off of you, you still have E to retreat. Flash brings it 3 even with 3. Other mids don't have such privileges, as they most likely lack even one mobility spell outside of flash.
For that Fizz part: if he used E to cross the wall, he couldn't kill you unless he was already fed. His E also doesnt move far enough to actually hit you with Trickster.
Meanwhile, assassins with only a single gap-closer is shut down hard. They cannot actually reach you again without you killing them first using Arise and Q. Two hits from one soldier equates to one nuke from another mage, and I don't think two attacks with one soldier is exceptionally difficult to pull off considering Azir's passive, his Q for repositioning soldiers and the slow on Q, and
Additionally, the more time wasted by an assassin is more time for your team to actually do something about him. Your ult also causes a chasm in the battlefield and makes it less likely for the assassin's team to assist the assassin in killing you.
As for positioning, you are Azir. You CONTROL how do you engage the enemy (notably, Shifting Sand's doubling of aggressive and defensive use). Your soldiers can be controlled from up to 1000 range away, and you can use Q to position them into a fight; there should be no reason you were "out of place" in a teamfight for an assassin to reach you with ease AND have your team too far away to assist you.
Burst assassin is a counter to everything not classified as a Tank or named Poppy, I don't know why you bring it up. If the assassin can kill Azir before he can react, that assassin would have rampaged over any other mid.
Therefore, in relation to other mid laners, Azir is an assassin counter. He has the peel and mobility to bring his mobility even with the assassin class, and he still has the safety and damage of a mage. He also counteracts against assassins better than most mid laners.