I just went through every single one of these troubleshooting steps outlined here with Apple Care on [my wife's] line, and still seeing this issue.
I have a sneaking suspicion that it's actually the SIM card not fitting tightly into the casing. When I took it out and popped it back in, it was much looser than I expected within the holder. There were a few times where it was showing "no sim", and I was able to get it to register the sim just by smacking the right side with the palm of my head, re-seating the sim card.
I used to product manage smartphones for Motorola back in the day, and I'm pretty confident that iPhones are manufactured to much tighter clearances than the sim card itself.
Regardless, I'm bringing the whole thing back to the Apple Store because I just don't want to deal with any more troubleshooting.