As we know, before we can use a mailbox it has to be approved by either Office 365 Global Administrator or by an Exchange Administrator.

When Server Side Synchronization was released only the Dynamics 365 System Administrator with the Global Administrator role was able to approve the mailboxes.

Later with the service update, the Dynamics 365 System Administrator having an Exchange admin role could also approve the mailboxes.

This requirement for mailbox approval can also be removed.

Navigate to the Administration > System Settings > Email tab and uncheck the Process emails option there.

For applying this change also, the Dynamics 365 Administrator needs to have either a Global Administrator or Exchange Admin role, else we will get the below error.

A failure occurred in CrmSecurity.

User should be a global administrator or exchange administrator to modify approval setting

We can reach the above setting from Settings > Email Configuration > Email Configuration Settings

We can also enable Verbose Logging, to get more details about the mailbox.

Download mailbox details can be used to download details about the mailbox

The Alerts section of the mailbox lists down any errors, warning, or information specific to that mailbox as shown below.

To monitor Server-Side Synchronization, we can also refer the below dashboards within the application.

Server-Side Synchronization Failures –

Server-Side Synchronization Monitoring –

Or Mailbox Usage Dashboard within Dataverse Analytics inside Power Platform Admin Center

https://docs.microsoft.com/en-us/power-platform/admin/analytics-common-data-service#mailbox-usage

More on troubleshooting and monitoring Server-side synchronization

Hope it helps..