Got it working and it did have to do with getting the phone properly connected through USB-C. Because I normally charge the phone using the MagSafe, the USB-C port on the phone wasn't cleared of all lint and debris when I was initially trying to connect it using continuity camera. Once I cleared the port thoroughly, it made the connection.
Oddly enough, the phone remained connected to continuity camera even after disconnecting the USB cord, so I assumed it had made the connection wirelessly (which it had), and that it would persist (which it did... for a while). Although it loses the wireless connectivity, it's always easy to restore continuity camera by plugging it into USB-C, but it's unclear why it, 1) establishes a wireless connection only to then, 2) lose that wireless connectivity and, 3) needs the hardwired connection to reestablish continuity camera.