It turns out that I ended up calling Apple. The initial advisor gave the usual party line about giving it some time to activate (24-48 hours). I asked if there was some issue with my cell phone number, and the advisor said no, and to give it 24-48 hours and how this process could be related to it being a new phone.
I then asked to speak to a senior advisor. The senior advisor looked and could see that my cell phone number was not checked but that the e-mail addresses were checked. And then a minute or two later, the cell phone number became checked.
This may be magical thinking on my part, but it's almost as if the act of looking at this setting on Apple's end triggered something, though the advisor states that he did not actually change any setting.
So it's theoretically possible that this would have happened on its own with enough time, but the timing of it related to the Apple call seems more than coincidental.
So now iMessages is working as it should with my cell phone number.