Update: I have replicated this on a couple of different saves. The spell count stalls out after a single fight.
Testing:
If I unbind, then rebind, I can go into a fight with a clean slate of zero spells cast. All the the spells cast in the next encounter will go towards the spell count. However any fights thereafter don't count the spells. So the item stalls out. I have only tested this with crucible fights and I'm curious if this is related to how the fights end ("event complete" vs going to a "loot bodies and back to normal" situation)
I finally leveled this item up by going into an encounter, getting a cipher to frequently cast ancestor's memory on my druid (refreshing abilities) and blasting spells until I reached the # of spells necessary.