This is correct. The volume alignment tools updates alignments3D/shift only, and not the location of the particle on the micrograph (location/center_x_frac and location/center_y_frac). The alignment shift is the residual shift required to centre a particle after it has been extracted and written to disk in the first extraction; so in cryoSPARC, the alignments3D/shift is effectively what defines the “particle origin” you mention. The geometric information needed by cryoSPARC to reconstruct a particle stack must be contained entirely in alignments3D, and since we can’t always assume that the original micrographs exist or have location information, we don’t attempt to do this (until the point of re-extraction with re-centering, if done).
The information displayed in inspect picks is location/center_{x,y}_frac, which is the pick location on the micrograph.
Re-extraction with re-centering updates location and alignments3D such that location stores the updated location of the particle on the micrograph, and the alignments3D/shift stores the small residual (<1px) shift value. E.g. in your example, this would cause the location field to now store the coordinates of the ASU on the micrograph, rather than the centre of each icosahedral particle.
In either case, re-extraction with re-centering is obviously preferred (especially if the new centre is an appreciable distance away from the original centre, as centered particles suffer less from the effects of CTF delocalization), but it is technically optional in this workflow. For small changes to the centre, you can run volume alignment tools with a new centre, and proceed immediately to local refinement and the geometry should be correct. The issue of inspect picks not showing the updated location is merely an artefact of this subtlety, and doesn’t affect anything else internal. Thus, the workflow you outlined in your first post (quoted below) should work.
Best,
Michael