Q: Face information not being restored

No, unfortunately the faces thumbnails are one of the things that are not stored in iCloud. See this document:

Use Photos and iCloud Photo Library on multiple Mac computers - Apple Support

Downloading again from iCloud to a new, empty library is like downloading on a different computer.

These items cannot be restored from iCloud:

• Books, cards, and calendars
• Slideshows
• Keyword shortcuts
• Unused keywords
• Last imported album (this album contains photos you most recently imported on a specific Mac)
• Faces tiles and Faces data

But iCloud stores searchable faces keywords. Renaming the faces will be easier than doing it the first time, because Photos will suggest similar faces.

Search for persons by name, name a few photos, quit Photos, to make it write the databases, then open it again and name a few more faces.  You should get many suggestions for additional faces of the persons you named, based on the tags that are still there.

You're welcome