Yes, you can absolutely buy the phone on release day with the discount. I got my Apple Watch on launch day last year via the military site with no issues. But you can’t preorder the phones, unfortunately. I’d imagine it has something to do with the financial side of things with each carrier and Citizens Bank, if trying to use the Upgrade Program.
edit: I’m talking about the preorder before the preorder, if that makes sense. I wish we could configure everything via the app with the discount like regular preorders so that all it takes is a few clicks and your phone is ordered.