See as the poll has over 200 votes now (and the ratios are unlikely to change by much) I think it's safe to say my own opinion now.
I think keeping his identity as a long range DPS mage is more important than the shuffle.
Shuffle is an iconic flashy play, but in all honesty, it's not even a very good one from a cost benefit perspective. You give up your ult, and your E, your two major forms of peel, to place yourself behind or in the middle of enemy lines, and knock them into your team. The risk is high, and the move is often not worth the value of being able to disengage an enemy trying to jump on top of you or your ADC in a SoloQ game.
I classify it has high risk, medium reward. Risk has to do with variance to me. How bad can this play possibly turn out vs how good it can turn out?
For you the shuffle it ranges from delivering your team's midlaner for free into the enemy team or winning the game. Very high risk.
How high is the reward? Well I consider it the average of the worst and best cases, so it's pretty much zero sum or average as far as rewards go.
Compare that to say an Orianna ult, which is low risk high reward play in my book. The worst thing that happens when you miss an Ori ult is "oops, lets try that again in 40 seconds" The average of the worse case scenario (nothing) and the best (winning the game) is pretty high overall.
If Azir is shifted to being more for engage he becomes a much more risky, less consistent champion imo. A few number changes in AP ratios is the difference between you shuffle a team and blowing them up, or you shuffle a team and dying before you get your DPS off. That I feel would continue to make balancing a headache. When he's strong he takes out your team with a shuffle, when he's weak die without dealing enough damage. This is way different from say lowering the ratio on say Orianna's W or R. You could cut it in half and it would still be a great ability because she's CC'ed their entire team still do AoE damage but most importantly she wouldn't be in the middle of their entire team!
Azir focused on being a long range sustain DPS is much easier to adjust and balance from a numbers perspective. His weakness becomes clear and he's treated the same way as an adc. Kill him quickly, don't let him draw out the fight and get a lot of value dealing sustained damage.