Azir gets outtraded early. That is why he tries to W AA Q AA to get TLD then back off. A big reason he has an "oppressive laning phase" imo is becuase people are too scared to trade with him early.
He doesn't have scary damage early. If you get even a little ahead of him goldwise he can't poke you without being in range for you to burst him.
I think a big thing is just that he's been out of the meta for too long, and barely anyone knows what to do around him.
And I can give you plenty of drawbacks.
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He has no hard CC save his ult, which means he is not one of the best mages to gank for pre 6.
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He has high cooldown spells early game that his damage depends on.
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Late game he is still squishy and cannot outtrade another carry without help or pulling a surprise shuffle from FoW because his lowered range puts him in the range of marksmen, who have crits and lifesteal while he lacks those traits.
Also don't separate poke, dps, and burst in your OP just to inflate your list and make it seem like Azir has more strengths than he really does.
I can name mages that burst faster than he can, can lock someone down, i.e. make picks better than he can without putting themselves at risk, who can sustain in lane better, who can poke better, siege better, etc.
I think you are just overhyping him to be honest.