I was using the the Share>Project Template from the ribbon, so now, after seeing there is a Consolidate Project tool, I tried that. I made sure "Package As Template" was checked, but the problem is the same. I think it has to do with the connection somehow NOT saving as UNC--even in my original project, the source for the database is shown as local path, with drive letter, rather than UNC.
I finally figured it out! In the window for Select Existing Geodatabase, type/paste the complete UNC from Windows Explorer into the Name field. Don't do anything with the browsing field at the top of the form; when you browse to the file geodatabase, even though it initially looks like a UNC path, if you click in the field you will see it replace the host name/sharename with the drive letter, and that is where the problem lies.
Where is this documented--how to get a UNC path for a file geodatabase connection into Pro? Am I the only one who can't find it, and spent a lot of time trying multiple different methods of trying to use a UNC path before finally hitting on the right one?