- Reduce the Azerite level requirement of the first ring to 1.
This would mean that the first ring is always unlocked. Since the majority of the Azerite armor's power is in the stats and first ring, it's pretty much always an upgrade. The inner rings can fill in as you upgrade your necklace, and give you small improvements as you play.
- Set the reforge cost to a flat 100g.
However, I think the hardcore are simply eating the cost of reforging, flipping between specs anyways, and then complaining about the costs on the forums. Setting the cost to 100g discourages casual reforging, but lets the hardcore be hardcore if they really want. 100g is cheap enough that an extra World Quest will cover the cost. This also simplifies reforging.
- Add a chance to get Azerite Armor when doing a Mythic+ of a given level for the first time each week.
So this basically gives you 3 chances per week to get Azerite Armor of a given tier. For example, do a +7, a +8, and a +9 to get i370 Azerite Armor. That's roughly comparable to a heroic raider's chance of getting a piece of Azerite Armor in a raid. But it keeps Azerite gear from being completely farmable and making Mythic+ mandatory. And since this is a separate roll on top of the current system, it doesn't reduce the amount of gear you get running Mythic+.
It also minorly encourages doing a few different levels of keys, which I think is probably a good thing for the Mythic+ system.
Those are the three changes I would have Blizzard implement. I don't think they help people who fundamentally oppose Azerite gear, but I do think they smooth out the most serious rough spots.