Check Focus Mode
Focus Modes (formerly Do Not Disturb) are a frequent cause for missing sounds, as they can override app-specific sound settings.
Look in the Control Center (top right of your screen) for a crescent moon icon or a colored icon representing a Focus.
If a Focus Mode is active (like Personal, Work, or Do Not Disturb), try turning it off.
If turning it off works, you can adjust the Focus settings to allow Mail notifications/sounds specifically. Go to System Settings > Focus and click on your active Focus. Check the Apps section to see if Mail is configured to be silent.
The Toggle-Off/Toggle-On Fix (The Classic Glitch Fix)
If the box is already checked, toggling it off and on again often clears the internal glitch.
In Mail Settings > General, uncheck "Play sounds for other mail actions."
Quit the Mail app (use Mail menu > Quit Mail or $\text{Command} + Q$).
Relaunch the Mail app.
Go back to Mail Settings > General and re-check "Play sounds for other mail actions."
Send a test email to see if the sound is back.
Check Mail App Sound Settings
The sent mail sound is tied to a general setting for "other mail actions," not the new message sound.
Open the Mail app.
Go to Mail menu > Settings (or Preferences on older macOS versions).
Click the General tab.
Ensure the box next to "Play sounds for other mail actions" is checked.
Check System Sound Settings and Output
If your Mac mini is playing other sounds through the Apple Studio Display, this is less likely to be the issue, but it's worth checking.
Go to System Settings.
Click Sound in the sidebar.
Under the Output tab, make sure your Apple Studio Display is selected as the output device.
Ensure the "Play user interface sound effects" checkbox is checked.
Check the main Output volume slider and make sure Mute is not checked.
Safe Mode and Restart
Sometimes, simply booting into Safe Mode and then restarting normally can clear various system caches that are causing anomalies.
Shut down your Mac mini completely.
Wait 10 seconds, then press and hold the power button until you see the startup options screen.
Select your startup disk (usually Macintosh HD), and then press and hold the Shift key while clicking "Continue in Safe Mode."
Once in Safe Mode, try sending an email (the sound might or might not work).
Restart your Mac normally (Apple menu > Restart).