I called the main CS number today, and after a 30 minute wait, I had an informative conversation where the agent explained that they have a 'new process' that started this week where they're trying to get all 'no progress' June orders set up with a delivery appointment. She alluded to all the existing problems with the process, but couldn't go into more detail.
She promptly put me through to the delivery team, who texted me out their direct phone/SMS number, assigned a DA, and promised to follow up later today if my DA hasn't reached out.
Both agents were happy that I called, and seemed motivated to help. Maybe they're starting to make some meaningful changes to help resolve some of these problems. Time will tell!