Bug or feature ?
After updating GMail on my Android 9 phone to version 2019.10.20.278647676, I go the the ugly message in the title, suggesting to remove the account (an "Exchange" account) and contact your system administrator (me ! 
I verified that our Zimbra server has an "A" in the SSL tests from https://www.ssllabs.com/ssltest/ , and searched the Internet and found many disappointed people with the same problem affecting Outlook-365 accounts (see https://support.google.com/mail/thread/19298229?hl=en ). All this activity is quite recent (november 2019).
I turned off support for TLSv1 and TLSv1.1, but to no avail. I could not find any documentation from Google about this.
So far the only workaround is to go back to the previous GMail version.
As a Zimbra admin in charge of a few thousands accounts this is a headache in the making, that can turn to nightmare very quickly. If any of you guys find a server-side solution I will be very grateful