They don't need any code reference if they have a written amendment. That's what they require here, and I think it's a good idea. I think they just got sick and tired of hacks installing a swivel bracket on to the drywall with no box.
I think they don't require it actually mounted, but they want to make sure whoever is wiring the place is very well aware that there needs to be a box on the finish
I just went to fix one about 2 weeks ago they had the fixture screwed to the drywall. Apparently the guy thought the inspector wouldn't pull the fixture off and on the final inspection, but he did
I've seen those swivels just as you described, also seen the sconces just mounted to the sheet rock or directly onto the siding and no boxes.
Just from experience coming in behind someone else that did rough in after the insulation and sheetrock was in and GC said there was suppose to be a wire there somewhere for the bath sconce. It wasn't where he thought it was and actually the end was buried in the spray foam insulation. So I can appreciate the idea that it must be terminated in a box before closing wall, but I can also appreciate that sometimes the HO cant make up their minds where they want the lights. And definitely don't want the circuit energized with a wire floating in a wall, seen that too, someone was going to put a light in and forgot.
Thanks for asking. He said that was alright with cat flex. His concern is the boxes.
He thinks the NEC says " No Renovation boxes allowed on new construction"
I told him to show me in the NEC or in local amendments. He got mad.
There are restrictions related to product used in new construction vs remodel, but not the one he is referencing. 1 is free floating (no staples) in wall with NM, allowed under code for renovation or fishing a new light in an existing space. Another those inline splice kits that originally developed in mobile home use that are now allowed to extend a branch line without a box, can only be used in an existing completed home but not on an open wall remodel.
To his point, an old work box is not prohibited, but how are you getting the cable secured to the framing member within the requirements of 334.30 of 12 inches. He wouldn't be able to verify that requirement be met after the wall closed.
Another might be thinking of the requirements in 406.5 "that boxes or assemblies be securely fastened in place....."
Both items if he is taking these as the reason it's being a bit picky, but it says what it says. And 406.5 has as part of it a caveat that unless otherwise permitted elsewhere in the code.