Well, here's bullet point saga of how I collected and even turned in some precursor items before starting much less even knowing this quest existed - keep in mind is at the time I didn't know what I was doing, only later after encountered issue and googled/researched and then in hindsight pieced together what happened.

1- Have ESO+, never bought any DLC - Clockwork was available whenever came out but like many was still catching up other areas so ignored CWC for awhile - accepted starter quest to go find CWC but sat ignored

2. Randomly puttering around Alikir delve, saw a shiny, clicked on shiny. Didn't know then, now know part of Precursor items. Found (3) precursor items this way but at time was mysterious misc thingy in my misc inventory tab

3. Fast forward some months, finally started CWC quest, got there, saw skyshard in the skybridge area, wanted skyshard, stumbled around trying to get up to that bridge. Found myself in what I now know is Assoc Zanon workshop.

4. In Assoc Zanon workshop, not knowing it is tied to The Precusor quest, saw cool looking clickable statue - I now know it was clickable because had (3) precursor items in my inventory - didn't know jack atm, just appeared clickable, so clicked - doesn't seem very controversial a player clicks on a clickable statue.

5. Whoosh - something happened. No idea at time - but now know the (3) unknown thingys I'd collected were taken and installed. Congrats player, you just skipped doing The Precursor quest and started the achievement hunt for installing all pieces and got credit for installing 3 of them

I want to say I am not -sure- this is what bugs it out. All I DO know is The Precursor is not obtainable a) at bulletin board spot, b) talking w. Assoc Zanon, or c) some skull that is supposed to be in Assoc Zanon workshop because it is no there.

Wanting to see if some way to get past this, went and completed all pieces of precursor achievement - handing them all in got me the combat dummy reward - great - AND offered the 2nd stage quest after The Precursor ---> The Oscillating Son.

Long story short, have completed every quest in zone, 23/24, including the 2nd stage quest that requires completion of the previous quest (The Precursor) but whether it was the before-quest actions or some other bug, however it happened, I'm stuck forever locked out from completing the DLC + the 50pt CCGA acheivement

Check out this website, https://lovelynorth.com/precursor-target-dummy/

Are you missing the Integral of Reason in the Machine District inside Brass Fortress?

Note: In order to enter the Machine District you must complete the story-line Clockwork City quests list below:

To the Clockwork City
In Search of a Sponsor
The Strangeness of Seht
Deepening Shadows
Lost in the Gloam – up to the point “Observe the Knight of Marrow’s Recitation”

Oh yes, it is a part to start assembling precursor without taking quest which bugged it out i presume. In my case I've read about the shiny pieces in my bag before as i was unable to offload them anywhere and was a bit nervous to destroy. So I came to the office all prepared demanding my piece of quest and a place to dump all those oily/rusty/shiny cogs and gears.

Go thru the quest list step by step to make sure you did not miss one. I thought as you did because I had parts before I got the quest. but that was not an issue. I had over looked a tiny little quest out by the Pavilion of Artifice. You pick up/read Durlocs handbook and then turn around and get her sword to return to her family in Ravenspire. Your mileage may vary.

List of all 25 quests
https://en.uesp.net/wiki/Online:Clockwork_City_Grand_Adventurer

One by the Pavilion of Artifice that I missed: https://en.uesp.net/wiki/Online:The_Merchant's_Heirloom