I have a suspicion that the root of your place problems is that you have created places that do not have a defined parent place. From the other thread you linked, I see the place in question here is called “Kaipatiki Creek Restoration site, Glenfield, Auckland, NZ”.
I can find it both on .org and .nz sites in the header search.
But I see that you cannot find it when searching for it in the context of a traditional project initiated on the .nz site.
For a traditional project place on the .org site, the options when you start typing “Kaipatiki” are numerous.
The solution here is to edit all of your places to assign an iNaturalist Standard Place as the place “parent”. Even though you gave it a name including a clear place hierarchy, iNaturalist needs to be explicitly told that this place is “descended” from a larger place.
In this case, you should select “Auckland, NZ” with the designation “state” (I know it’s technically a region, sorry about that), as indicated in the screen shot.
Once you add the Auckland parent place for all your Kaipatiki Creek places (maybe some can be deleted too?), your place searching problems should be resolved because they will all be recognized as clear descendants of the New Zealand standard place associated with iNaturalist.nz.