Switching to the staging branch did not fix the issue. For further information, the Pi is a Pi 3 Model B+.
To re-establish the symptoms:
Booting the Pi 3 with the latest January 2023 OSMC image, out of the box (stable branch being default) will output through composite video. The splash screen appears.
After the splash screen appears, Kodi does not start, and a black screen is shown indefinitely. sdtv_mode and hdmi_ignore_hotplug make no changes as output is working, but Kodi is not launching. I am unaware of any extra changes that need to be made to the config file to enable anything like video drivers to enable Kodi to launch properly on a Pi 3 B+. I have even checked for the dtoverlay settings and cmdline settings for composite video. No effect.
Strangely, while not related to OSMC, I installed LibreElec on a different SD card and used the cmdline and config parameters and Kodi launches on that distro, which is great, but it’s not OSMC. I at least know there are no issues with my Pi at all, and I highly doubt the SD card has issues as I’ve reflashed it multiple times.