I am running the Big Sur (11.1) OS on my MBP and recently encountered a version of this problem: Mac Mail failed to sync reliably with my work Exchange account. In order to force a sync I would have to quit and restart the Mac Mail app, and sometimes even to reboot or at least to log out and log in again. Frustrating.
After reviewing this thread (thanks to all who are contributing) I seem to have resolved the problem.
First an observation: This problem emerged for me after I had recently had a unusually high number of incoming emails in a single day. I don't know if this was a coincidence or not, but I suspect that it may have pushed the total number of messages in my inbox over some threshold limit.
A first failed attempt: Operating with the hypothesis (drawn from this thread) that the problem might related to volume of emails, I first tried archiving thousands of emails from my Inbox by using the Mac Mail application. This did not solve the problem, perhaps because the continuing sync failure left those messages in the inbox on the Exchange server.
As to my remedy: Looking for an alternative, I used the Microsoft Outlook for Mac app (not the web-based Outlook interface) to access my Outlook inbox. I scrolled to the very bottom (i.e., oldest) messages in my inbox and deleted several hundred. Interestingly, these were messages that I had moved out of my Mac Mail inbox some months ago, but they had not been auto-deleted from the host server inbox.
I then rebooted, and the problem seems to have been resolved. I am now back to reliable, "real-time" syncing. Whew.
I am not sure whether my actions actually fixed the problem, or if this was just a happy coincidence. (A number of contributors to this thread have noted that the problem is intermittent.) But I offer my experience as a data point in the larger collective effort to understand this problem.