Set up a row of 7 kegs, put potatoes in them.
Set a chest at one end with 5 more potatoes, Profit! (or not)

Saw Pam count those 7 as being 'kegged', Saw 7 potato juice produced and collected into box,
Final 5 juice popped into the box, but Pam didn't count those as ever being kegged.

Does Automate not work with quests?

(I am only a farm hand in our multiplayer farm - if the host has access to save files... I'll research how to retrieve a save file - are there save files from 12 days ago when I took the quest? Yes, I see they ought to be in SMAPI. I'll see if I can replicate.)

I noticed this with the same quest. The quest updated upon taking the items out of the chest.

Hi! Special orders like that generally count items received in your inventory for the first time while the quest is active. So they'll be counted when you collect them from the chest, as xeropresence mentioned.

I believe i've found a bug related to this. I reloaded my save to confirm.

Items are counted towards the quest for the first time you put them in your inventory, but new additions to the stack do not.
Reproduce it like so:

Start the Strong Stuff quest or similar.
Start 6 kegs one day. Start the other 6 the next.

Collect the 6. It counts towards the quest as expected. Put them back into the chest.
When the other 6 finish, collect all 12 and it does not count the additional 6 towards the quest tally.

I encountered this accidentally as I had no other chest I wanted to store the product. So I put them back into the chest.
I am fortunate I have a mod that lets me roll back to previous days. Otherwise, I'd have failed the quest.

EDIT: I've posted the bug on Nexus rather than update this closed issue or create a new one as per the recommendations.