Hi Kara,
you are correct.
When you arrive from auckland on AA, you won’t clear immigration at T4. Rather you’ll take a 10 min long walk that takes you all the way from your arrival gate in T4 to the TBIT immigration lines through the connector and from there you can use a global entry kiosk to get out quickly. With AA it’s convenient that they are connected to TBIT with a short walk. Hope this helps! Cheers!