A dry stone wall (like any other dry laid structure) will likely fall down in a short time if the one who builds it lacks integrity, because integrity is something that needs to be built into a wall. Whether they utilize stones that go all the way though is not the issue.
The word 'through' is closely related to the word 'thorough'. (In fact until early modern English the two words weren't even clearly differentiated).
The meaning had to do with being completely and entirely done from 'end to end' and 'side to side'
To thoroughly stone a wall is to have its length and depth made entirely of good workmanship. If the person building it is being as thorough as 'possible' it will be done right. It is quite possible to do a proper job 'throughout' even if throughstone material is scarce or missing .
By contrast seeing a wall that obviously has throughstones in it, doesn't imply that the builder has done a 'thorough' job. (Nor does the fact they've used a lot of them mean they've been somehow more thorough)
The point is - all the stones in a wall are 'through stones', if the wall has indeed been built 'thoroughly'.
I've seen enough evidence to conclude that the walls a scoundrel or a blaggard builds, no matter what their credentials, or how many throughstones they use, will always lack integrity.