General recommendation – whenever it’s possible to block access via user capability – remove that capability from the user role. Use ‘Admin menu access’ module just in case you have to leave the capability in the role to provide access to some other menu items.

Links menu is protected by ‘manage_links’ capability. I suppose it is not used anywhere else. So remove it from the ‘production’ role.

Btw, I successfully blocked ‘Links’ menu as ‘Not Selected’ at recent test with URE Pro v. 4.24.6. It just confirms your guess that discussed site has something special…