Thanks for responding, Mr Finch, but no that's not the issue.
- This isn't an early iPhone 6: I bought it less than a month ago on May 10 from the Apple Store directly and it came with no SIM at all since it is the unlocked, SIM-free version
- AT&T's rules on VoLTE-compatible SIMs are well-documented on their support site, and the SIM they gave me at the AT&T store when I took my unlocked SIM-free iPhone to them conforms to these rules (there are two digits -- the 8th and 9th I think? -- that have to be specific numbers)
- AT&T tried replacing my SIM anyway on Sunday, without effect.
My issue (and probably that of the original poster) has to do with AT&T's provisioning, not the SIM. When Apple first came out with the SIM-free unlocked model, which is a1586 (the model that was originally only sold domestically as a locked Sprint phone, as opposed to a1549, which was sold from day one for AT&T, Verizon, and T-Mobile), everybody who bought it reported problems with VoLTE on AT&T, Verizon, and T-Mobile. That was in January.
By March, I started seeing people saying that they had got AT&T (or Verizon or whoever) to finally admit that a1586 is capable of VoLTE on their systems and provisioning their accounts. Unfortunately, naturally, none of those people have posted their case #s on line, so I can't cite them when speaking to AT&T customer support. So far none of those who have had their accounts successfully provisioned have responded to my requests for information either as yet.
All of these people have said that it took them a while to get to the highest level of AT&T (or Verizon, etc.) tech support ("Tier 3") and only then were they able to speak with someone who had the correct access level to override the system and enable VoLTE in their accounts. This is all because Apple sold the a1549 model for those three carriers initially, but the unlocked one is a different model.
