MS SQL 2012 AD Login Frustration

I have an MS SQL 2012 server that is configured for mixed authentication. Originally, a number of AD user logins were added directly and working. I wanted to make this a little easier to manage, so I did the following:

1. On SQL server: Deleted the logins that were linked to AD users.
   
2. In AD: Created a security group for "This Server's SQL Admins"
3. In AD: Added the appropriate users to the "This Server's SQL Admins" group.
4. On SQL server: Created a login for the group. The Login name is "domain\This Server's SQL Admins". Serer Roles are public and sysadmin, no user mappings, Secureables are the SQL Server with a grant for "Connect SQL", status is grant permission to connect to database engine and login enabled.

I have the same setup on another MS SQL 2012 server that is working successfully. However, in this case, it is not working. If I add the...