Sql-server – BUILTIN\Administrators accidently removed

This is expected. LUA strips administrators membership from your token. From Getting Started with User Account Control on Windows Vista:

To help prevent malware silent installation and computer-wide infection, Microsoft developed the UAC feature for Windows Vista. Unlike previous versions of Windows, when an administrator logs on to a Windows Vista computer, the user’s full administrator access token is split into two access tokens: a full administrator access token and a standard user access token. During the logon process, authorization and access control components that identify an administrator are removed or disabled, resulting in a standard user access token. The standard user access token is then used to launch the desktop, the Explorer.exe process. Because all applications inherit their access control data from the initial launch of the desktop, they all run as a standard user as well. Contrasting with this process, when a standard user logs on, only a standard user access token is created. This standard user access token is then used to launch the desktop.

Since your local desktop token has the administrator membership removed, it cannot log in locally to SQL Server if you expect to log in through builtin\Administrators. On remote access your token preserves the Administrator membership and you gain access.

The solution is to grant SQL Server login rights through an explicit domain group account and add all users that require access to your SQL instance to this domain group. Do not rely on Administrators membership, is a bad practice.

Per BOL: Configuring Windows Service Accounts and Permissions

SQL Server Agent Login and Privileges
The per-service SID of the SQL Server Agent service is provisioned as a Database Engine login. The per-service SID login is a member of the sysadmin fixed server role.