Omigod! What a tortured phone call you had.
I contacted chat yesterday (from MyATT, contact us, then select chat) and explained all clearly. WiFi calling working. Continuity working. Between an iPhone 6s Plus as the main to any of my other devices (iPad Pro 9.7, 3 Macs). All good. NumberSync NOT working with any, even though it had been working since original launch. Stopped working last Monday or Tuesday,
They checked to make sure HD Voice was provisioned, since this is a requirement. It was. They asked me to reboot the phone. I did. They also asked me to check each of the devices to ensure they were all logged into the same iCloud account. They were. Lastly, the asked me to check to ensure caller ID on all the devices was set to my phone number. All were. These were very reasonable requests that are required for NumberSync to work, and showed that they knew what they were talking about. I explained that I'd been patient waiting for them to fix, But it had been too long.
They issued a case (or ticket) number and promised they would work on it.
I found using chat to be much easier than phoning because I could lay out the case clearly, once, and they could refer back to it. I was also very lucky to get a representative who seemed to understand what I was saying -- and the technology -- and understood that I knew the nuances of how WiFi calling, continuity, and NumberSync all worked, how the phone had to be provisioned, and didn't once blame Apple or say it was a FaceTime problem.
I checked back with chat today, just giving them my case number rather than describing the whole problem all over again. They told me the expected resolution date was 8/31. I said that seemed like a very long time, especially since the problem had been going on a week. They said they would try to put an expedite on it, and that I should expect to hear back if it was fixed or if they needed more information.
Yes, I have successfully used NumberSync to initiate and receive calls with my iPhone off or disconnected from WiFi, just the same way I have done with my iPad. My call detail shows I make more NumberSync calls than WiFi Calling calls, and no too many cellular calls. I often leave my phone in a different part of the house or office when I'm working on the iPad or on one of the Macs now that I have that freedom.
I should note that my iPad does have a cellular modem, but I know (as you do) that NumberSync works with or without it. Clearly my Macs have no cellular modems and can use NumberSync just fine. The only thing the Macs and non-cellular iPads CAN'T do is use NumberSync over cellular data. (Not very useful, since phone calls would use up data allotment anyway.)
It's amusing (and I'm sure very frustrating for you) to hear customer service agents be so sure they are right when they clearly aren't. You are a saint for staying so calm.
I poked around on the Mac earlier today, as I noted above. The error messages in the console log (overall, and especially in the coretelephony section) show that, as long as the iPhone is set up to allow the Mac to do NumberSync, and as long as FaceTime on the Mac is properly set up, the Mac tries continuously (maybe every minute or two) to make a connection to AT&T and fails. It's looking for some files it doesn't have, and not getting tokens it should. Once I turn FaceTime off on the Mac, all these errors stop and it stops trying. I imagine the iPad is going through a very similar cycle. (I could try to figure out how to get the logs from the iPad, but I'm not sure I'm motivated to take the time.)
It would be nice if AT&T acknowledged the problem, but I surely don't expect them to do that. I'm tempted to install the most recent public beta of iOS 10 on the iPad or macOS Sierra 10.12 on one of the Macs to see whether anything is different, but I'm a bit too busy right now to take the time to do it.
Whatever the issue is, it's very specific to FaceTime. WiFi Calling and the MCell that my mom has and that some others with older iPhones in my company use haven't been impacted at all. AT&T doesn't seem to have lost their vowifi (or voip) capabilities broadly. If I had to hazard a wild guess, I would imagine that some authorization servers that issue tokens or oversees encryption that interacts with FaceTime are not working properly. That might be why TMo and Sprint customers aren't complaining, which they would be if Apple's servers (which must be involved at least to authenticate, start, and monitor connections since much goes on through iCloud as far as knowing which devices can be or are connected) were the problem.
Anyway, I'm guessing this will all be cleared up pretty soon. Who knows -- when it comes back on line as before, it may even gain in dependability.