One astute poster detailed the handshaking between the iCloud mail server and the macOS and exactly what handshaking gets stuck. Both macOS Mail App and iCloud mail server are waiting for the other to send a tag -- effectively stopping the self-synchronization between the iCloud mail server and the macOS app. This stand-off does not exist in the iOS version of Mail. Still exists at least on Catalina if not Big Sur. Any way to get this to the Apple Mail macOS development and maintenance teams?
I wish there was a better way to bubble up really serious user experience issues to Community Specialists and Apple Mail team. From what I can tell, several thousand users have the same problem with Apple Mail running macOS on their desktop. The common user experience is:
a) macOS on desktop, Apple Mail application
b) hangs forever with "Moving Messages 1 of xxx" in bottom left corner of Inbox screen (I have 580 for xxx and growing.)
c) inbox INCLUDES iCloud mail account (IMAP)
d) new messages delayed for hours or until Apple Mail re-launched
e) attachments not downloading for new messages (either blank email with "loading..." displayed or the email shown with the icon for the attachments shown with blue "Downloading..." under each attachment icon) -- relaunching Apple Mail does not help
f) at the same time, there is NO problem with the mail on iOS devices -- people running to their iPads and iPhones to salvage their mail and workday