I decided to actually try it out again, but... I noticed pressing it only changes the button back to "Start Manual Override" before refreshing the page, which also reverts the button to "Menu-based Order Entry". So pressing it once doesn't really do anything besides refresh the page.
Quickly pressing the button twice, or rather pressing "Start Manual Override" before the page finishes reloading, does open the Manual Override input form.
It works the same way on my phone. I'm using Chrome on both my PC and my phone, so it's possible it works on Firefox etc.
I didn't dare to actually input anything as "Cancel Manual Override" disappears once the page reloads too.
Thank you for your time again!