OK, here is a clue:
after reinstalling iTunes 12.5.3, I noticed that it wasn't syncing contacts and calendars, and saw that Settings / Info tab shows: "Microsoft Outlook must be installed in order to sync calendars." I already have Microsoft Outlook 2013 installed.
Scratching around, I found the following solution: exit iTunes and rename the sync services folder in Windows from:
C:\Users\<my account>\AppData\Roaming\Apple Computer\SyncServices
to
C:\Users\<my account>\AppData\Roaming\Apple Computer\SyncServices_Old
Once it has been restarted, this forces iTunes to recreate the folder and repopulate it with fresh data. This worked. The folder was rebuilt (after a few minutes).
However, when I ran the sync again, Outlook threw the following error:
"Cannot start Microsoft outlook. Cannot open the Outlook window. The set of folders cannot be opened. The information store cannot be opened."
This led me to another solution, i.e. to close iTunes, and Outlook. To sign out from and then uninstall iCloud. Reinstall a fresh iCloud from Apple. Restart PC. Sign in to iCloud again. Then restart Outlook.
After doing all of this, the iPhone 6+ is syncing smoothly again!
Bottom line, maybe the problem lies in a corrupted SyncServices folder? I suspect that the renaming of SyncServices broke the iCloud integration with Outlook, but hey, what do I know?