Recently acquired all of the Heart of Azeroth essences from Mechagon the outdoor zone, notably all 3 ranks of the Purification Protocol, and they're not registered as completed, and I have no way of acquiring them again to try to refresh the addon while they're in inventory as the game will not allow me to craft another or buy more.

Hmmm, you may need to open the Main ATT list /att since Azerite Essences don't have a 'collection' event sent to the game client, the only way for ATT to check they are collected is when the Update logic checks the Sourced data (which happens for everything when opening the Main list).

Give that a try and see if they get found :)

It actually seems to have resolved itself today when I logged in, though it has been failing to update for several days. The only thing different was the Heart was actually in my bags when I logged in, instead of stuffed back in the bank. Or perhaps that I carried it with me to BFA content, where the Heart menu lights up instead of being grayed out as it has been the past few days. The /att command seems to have detected several other things though and increased my overall percentage by a tiny bit, so will definitely use that again for other issues.

Edit: I couldn't let it go and kept looking, discovered on the Wowhead page for the Heart that there is sometimes a bug where it registers as dormant everywhere, when it's only meant to be dormant in Shadowlands. It can be debugged by relogging in a zone where the Heart is meant to be functional. So the bug with the Heart was clearly the real issue, in case anyone encounters this later on. It had been nonfunctional for possibly a few weeks, but I am uncertain how long exactly, and I thought it had been patched to that state.