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THey are also used for crafting at the Augery station and some of the potion crafting.
Edit: worth mentioning that despite the gnosis cost for making apprentices, they pay for themselves (and then some) between the leveling of relics and the gnosis shards earned for their skill increases, especially once you have the relic that boosts that even more, so don't let the gnosis cost dissuade you from doing that.
then look for relics that do "get x more on level up" and also you can pulverize unused relics that gives shards based on the relic's level.
What did you think you were asking here, if not in the words of your own title "What uses gnosis shards?" Maybe you didn't need the first paragraph since you ought to be aware when you spend them in rituals (but that's risky to assume), but then the second paragraph finishes the topic off.
As best I can tell, the base income for learning is just one shard per skill level? And an apprentice only gains at most four total skill level cap. That's nowhere near paying off the cost of apprenticeship.
Of course, as you say, you can boost that, depending on how hard you work at it.
There's the displayed relic type that gives you a chance of extra shards on learning, IIRC the ones I have are around +1 shard per skill, though of course that's more useful with initiates picking up their many intro skills.
Then there's element boost relics, which around level 7 start giving +shards on learning or teaching the subject. +2 per level for the ones I've got, though it might improve at higher levels. If you have enough of those for your apprentice to have one, that's great. If you've also got per-element teaching specialists with the same boost, it should stack...
Even with all that though, apprenticeship fees would seem to utterly swamp the direct gnosis returns of apprenticeship. Picking up something like 20 shards if you do everything possible? So to actually break even you'd need to be pulverizing a relic.
Of course, a graduated apprentice produces two relics in their career and feeds a lot of experience into 3-5 relics (depending on build and whether they're gifted) so aiming to pulverize one relic per apprentice, amortized, probably is feasible and pays their bills?
- Apprenticing, Hiring, and all types of Advanced summoning rituals
- Sal Alembroth (1) and Sal Oblivium (4), the 'potions' used for pre-battle scouting
- Crystal Portents (3) and Crystal Prophecies (10), used for pre-battle weakness changing
I'm pretty sure that's really it.
Thus, if you have a constant drain on your Gnosis shards, you probably have a large, possibly open-ended, order of one of the pre-battle items. Check your alchemy and divining workrooms.
Note that people provided this answer before getting into the other things you're complaining about - the first and fifth posts both say this, though you've provided a more detailed description of the alchemy and augury sinks.