Greetings,
Ensuring the notifications for the Mail app are working as expected is important. To confirm, have you verified the notification settings as outlined in the following article? If not, verifying the settings can help with the unread email notification bubble. You can also find these steps listed below.
- Click the Apple menu towards the top-right of the display.
- Click System Preferences in the menu.
- Click Notifications & Focus.
- Then click Notifications.
Change Notifications preferences on Mac - Apple Support
In addition to verifying the settings, when closing the Mail app are you hitting the read circle in the Mail window or are you quitting the application entirely? If you quit the application, this would be why the notification bubble isn't updating. If you would like to provide feedback about the notification settings, you can do so via the Product Feedback website. This would ensure the appropriate individuals receive and review the feedback you provide.
Take care!