Hi Fred,
So you synchronize the new site from the TSC3 to the PC sync folder, then open a project with no job site, and then run Job Site Manager > Add external site?
Do you have more than one controller in the sync folder with this job site, and they have different site cal or control points? (or they are absent) If so, it won't automatically import the site-level data. (Or at least it didn't back in the day.)
There are project settings that prevent the files in the sync folder from being overwritten:
Project Settings > Computations > Field Data > "Protect site calibration files" and "Protect control point files"
But that shouldn't prevent the site cal + control points from being added to the VCE project.
When you add the external site, does the Job Site Manager have Resolve buttons (red dot) since your controller has site cal and control points, but the VCE job site doesn't?
You should be able to import your coordinate system, site cal, and control points that way instead of having to find the folder and drag-n-drop. This will also push out the site cal + control points to the other assigned controllers in the sync folder.