What you describe is almost certainly due to someone creating a DDL trigger which requires xp_cmdshell.
Adding or Deleting a Login Fails with xp_cmdshell Error
I have such a weird issue. I have a brand new SQL2019 FCI with CU9 and I can't add or delete logins without having xp_cmdshell enabled. If it isn't I get this error when trying to delete/add/disable etc a login:-
SQL Server blocked access to procedure 'sys.xp_cmdshell' of component 'xp_cmdshell' because this component is turned off as part of the security configuration for this server. A system administrator can enable the use of 'xp_cmdshell' by using sp_configure. For more information about enabling 'xp_cmdshell', search for 'xp_cmdshell' in SQL Server Books Online.
Why does login admin activity need xp_cmdshell enabled?