This could probably go on forever, depending on your point of view, but here is my take on this.
I think the reason you don’t find an outright requirement in the NEC itself is because of the way the NEC defines a panelboard in Article 100. Using this component definition, the panel boards themselves would not be required to have a cover, especially when accessible only to qualified persons.
However, the "enclosure" that the panelboard is mounted in is designed in compliance with various standards, such as those in NEMA standards Publication 250 and NEMA standards Publication PB-2006, ANSI/UL 50, UL 589, and I'm sure, probably several others if we dig deep enough.
If I were the inspector on that job, I don’t think it would be a stretch to argue that if panelboard itself is listed and designed to be factory or field mounted in a NEMA rated "enclosure", then it must be listed as an equipment assembly, and that the listing for the equipment assembly with the NEMA rated "enclosure" includes a cover, regardless of whether or not the location is accessible to only qualified persons, since it is not possible to achieve even a NEMA 1 rating without a covered "enclosure".
I would probably site 110.2, 110.3[A], 110.3[B] before even downing my first morning cup of coffee, but that's just me.