It seems as though raspberrypi-firmware-tools-20160703-1 does not provide libbcm_host.so that kodi-rbp-16.1-2 expects it. Kodi does not launch with it installed. Output from journalctl:
$this->bbcode_second_pass_code('', 'kodi-standalone[287]: /usr/lib/kodi/kodi.bin: error while loading shared libraries: libbcm_host.so: cannot open shared object file: No such file or directory')
Downgrading to 20160627-1 fixes the issue.
As an aside, is the forum still the best place to report bugs like this?