This has nothing to do with what I believe, it is what Apple has clearly and unambiguously stated:
Capture photos that come alive when you touch them. Then you can pick a different key photo, add a fun effect, edit your Live Photo, and share with your family and friends. All from your iPhone 6s and later.
From the article I posted above: Take and edit Live Photos - Apple Support
Also, you don't need a 6s or above to see Live Photos. I can see them on my iPhone 5s.
The screenshot you provided to show that the Live Photos button was not showing had the camera in the Square mode setting. So, that screenshot doesn't show anything since that button doesn't ever show up on that setting.
Your best bet at this point is to take the device to your nearest Apple Store. There is really nothing more I can provide remotely. I have provided the facts, and in this case, that is really all that I can do from here.
Best of luck,
GB