You can't target the gear you want at all.
There are 10 M+ dungeons, all of which drop ~10 different items per dungeon, and are all eligible to be looted from the cache (the only way to actually get the gear). This translates to a 1/100 chance to get the piece that you want, but that's only one piece. Then you've got to get 2 more pieces, each on separate 1/100 "drop" chance. Let's be lenient here and say we are given 6 months per content patch. This means that the probability of getting the items we want is actually orders of magnitude lower than the probability of getting the ideal legendary items in Legion.
The vast majority of traits are useless and uninteresting.
When all of our abilities were stripped away in 8.0, we were promised Azerite armor that would bring back the gameplay changes that artifacts brought. There are very few circumstances where this is the case, and most of the top-performing traits are just stat-padding, providing no progression or gameplay enhancements.
When we get "upgrades" we have to re-farm the same traits that we have already farmed.
We complained about having to farm traits on Legion Artifacts, and Blizzard's response was to give us only 1 or 2 traits, but to consistently lose them and have to re-farm them for the whole expansion. How is this an improvement?
Every single gear change or upgrade requires simming.
At no other point in WoW's history has simming been so required to find out if an item or trait is actually an upgrade. The proc rates, damage bonuses, or other interactions are so unintuitive that we have absolutely no idea what is an upgrade or not. At no other point in WoW's history has the casual player been at such a disadvantage for being uninformed.
340 traits are out-performing 385 traits. Significantly.
This is just embarassing, and should not be the case. How did this sort of imbalance between traits make it to live?
The reforge cost re-introduces the exact spec-lock problem people had in Legion, but worse.
Oh you want to PvP? Gotta reforge your Azerite! You want to PvE now? Reforge time! Oh, you need to tank this boss? Hearth and reforge your Azerite! People are getting their reforge costs into the thousands of gold, and it doesn't go down fast enough to allow us to reasonably perform multiple roles. How was this supposed to improve gameplay?
The massive swings in power during "tuning" passes mean we have to hold onto all gear.
We have no idea what is going to be our best trait next week, and what is best today may just be completely gutted tomorrow. Our only option as a player is hold onto every single piece of Azerite Armor and hope we have the right ones when Blizzard swings power in a different direction. Not only that, but when new traits are introduced to replace old traits, per Ion's comment, we are likely to encounter exactly the same issue.