After playing around for far too long I had the inspiration to run SQL Profiler, capturing all errors (don't know why I didn't think of that earlier!).
During the Send Test Mail process it revealed that
The EXECUTE permission was denied on the object 'sysmail_help_configure_sp', database 'msdb', schema 'dbo'. against System\NT AUTHORITY
So I added a bunch of permissions to that user, and suddenly everything works!
My NT AUTHORITY user now has following permissions in the msdb database:
- Member of the DatabaseMailUserRole
EXECUTE permission for the following:
- dbo.sp_ProcessResponse
- dbo.sp_readrequest
- dbo.sysmail_configure_sp
- dbo.sysmail_help_account_sp
- dbo.sysmail_help_admin_account_sp
- dbo.sysmail_help_configure_sp
- dbo.sysmail_help_configure_value_sp
- dbo.sysmail_help_principalprofile_sp
- dbo.sysmail_help_profile_sp
- dbo.sysmail_help_profileaccount_sp
- dbo.sysmail_help_profileaccount_sp
- dbo.sysmail_help_profileaccount_sp
- dbo.sysmail_help_profileaccount_sp