Some perhaps new info from my visit to the Durham, NC Apple Store:
The Genius I worked with (Alvin) was EXTREMELY helpful; lucky to get him as he noted iPhones are of interest to him and his in-store specialty. Alvin said this was a rare event but one he had seen a couple of times before. According to Alvin, the issue occurs when a CDMA iPhone 4 is set to be unlocked. Recall that Verizon reached an agreement during a spectrum auction a few years ago to leave their phones unlocked for world-band use as a condition of getting more LTE spectrum. According to Alvin, occasionally an iPhone 4 gets registered in "the system" (not sure if this is Apple or VZ system) as being in an unlocked state.
Of course, a CDMA iPhone can't be "unlocked" -- locked/unlocked only applies to GSM phones. As a result, when you try to go through phone setup, the phone's software awaits a SIM card, which it doesn't have. (That is to say, because the network database has a certain flag for unlock status, the phone believes it is GSM and the software's logic looks for a SIM card.)
Alvin took down my MEID and other info and escalated a case. He said the solution was to have the unlock flag flipped on my account and an activation message sent to my phone, if I understand correctly. I was told to expect a call in 24-48 hours.
In my excitement and in a hurry to make it back to a work meeting, I didn't remember to get a case number, but can do so when I get the call back.
Alvin said this was rare, something he saw perhaps 4 times out of what he estimated to be 2,500 or so cases he has worked on since the fall. I do wonder if something was pushed to phones recently (to Meg's point of a recent carrier update) that caused more phones than usual to go into this unusual state.
I can't vouch that this is the solution for sure but am hopeful that it is. I was relieved that Alvin didn't force me through another DFU, restore mode, etc.; based on my description and showing him the thread he knew what this was. So, hopefully this is progress.
FWIW, my iPhone 4 is black, not white; model A1349.