Two quick updates:
Based on reports, we were able to find a strange behavior where a player's loot history could be inaccurate depending on the loot specialization they had selected. We believe this is why some players with average item levels over a certain Azerite threshold were seeing rewards lower than expected (e.g. players with 371+ average item level getting 355 Azerite caches). We have a hotfix currently in testing to correct this issue.
Secondly, we agree that the requirements for the 370 caches were a little too high initially. With things like this, we tend to err on the side of being too conservative so that we can relax restrictions later if need be, and that's what we're doing here. We're going to lower the average item level threshold required to see 370 Azerite Armor in Emissary caches slightly in another upcoming hotfix.
Both hotfixes are being worked on now, with the goal of getting them out later today.
Why does it even need to be based on your item history to begin with? Why can't it just be based on your general item level like every other world quest and emissary gear reward?