In SQL Server Mgmt Studio, why don't triggers automatically get updated when tables get renamed?
Trying the following in SSMS..

CREATE TABLE [dbo].[T]
  (
     X INT
  );

GO

CREATE TRIGGER [dbo].[TR]
ON [dbo].[T]
FOR INSERT
AS
    SELECT *
    FROM   [dbo].[T] 


Then renaming the table as T2 and scripting out the trigger returns the following so I assume you are referring to the second table reference.

CREATE TRIGGER [dbo].[TR]
ON [dbo].[T2]
FOR INSERT
AS
    SELECT *
    FROM   [dbo].[T] 


If you use SQL Server Data Tools to manage your database schema instead of making schema changes directly in SSMS then this does have a rename refactoring. It isn't 100% comprehensive as you would still need to review for any possible usages in dynamic SQL or references to the table in external scripts but it would fix the issue above.