SQL Server Installation – Is SQL Server Agent Enabled by Default?

Any idea why the access is denied if the batch file is in desktop?

Yes. Because it is a rotten idea. The service agent account has no rights on your personal folders and your personal folders include the desktop. And you should keep it that way.

Your personal fodlers are btw., locked also for the admin accounts etc.

If you need sql agent to do something, I strongly suggest you agree on a location for that to happen and move the batch there.

There are couple of steps that you can do to address the problem that you faced :

1. Have a startup stored procedure using sp_procoption that runs and checks and alerts you that SQL Agent is not running.

   IF EXISTS (  SELECT 1 
          FROM MASTER.dbo.sysprocesses 
          WHERE program_name = N'SQLAgent - Generic Refresher')
   BEGIN
    SELECT @@SERVERNAME AS 'InstanceName', 1 AS 'SQLServerAgentRunning'
   END 
   ELSE 
   BEGIN
   EXEC msdb.dbo.sp_send_dbmail
   @profile_name = 'yourDBMailProfile',
   @recipients = 'dbateam@somecompany.com',
   @body = 'Please check the status of SQL Agent. It is not running !',
   @query = 'SELECT @@SERVERNAME AS [InstanceName], 0 AS [SQLServerAgentRunning]',
   @subject = 'SQL Agent is not running',
   @attach_query_result_as_file = 0 ; -- set it to 1 to receive as txt file attachment
   END
   

2. Set SQL Agent service to restart