Looks like I spoke too soon. Although after running the commands posted by mscott_mdm above, I was able to see all of my enrolled devices in Profile Manager again, upon inspection of the logs after issuing the command to update all devices, I get tons of log entries in the Profile Manager Device Server log that look like this:
1::Oct 29 13:40:25.241 [90019] <10.0.10.122> {LogElapsedTime (common.php:74)} Time since script start: 214us [https://server_url_obfuscated/devicemanagement/api/device/checkin]
1::Oct 29 13:40:25.241 [90019] <10.0.10.122> {require_once (mdm_checkin.php:11)} vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv - PUT checkin
0::Oct 29 13:40:25.241 [90019] <10.0.10.122> {require_once (checkin.php:13)} checkin: 'TokenUpdate'
0::Oct 29 13:40:25.242 [90019] <10.0.10.122> {LogException (common.php:426)} EXCEPTION: 403 Forbidden - old enrollment scheme not supported at
0::Oct 29 13:40:25.242 [90019] <10.0.10.122> #0 /Applications/Server.app/Contents/ServerRoot/usr/share/devicemgr/backend/php/ta rget.php(204): DieForbidden('old enrollment ...')
0::Oct 29 13:40:25.242 [90019] <10.0.10.122> #1 /Applications/Server.app/Contents/ServerRoot/usr/share/devicemgr/backend/php/md m_checkin.php(61): Target_for_checkin_request(Array, true)
0::Oct 29 13:40:25.242 [90019] <10.0.10.122> #2 /Applications/Server.app/Contents/ServerRoot/usr/share/devicemgr/backend/php/db .php(393): _checkin_transaction(Array)
0::Oct 29 13:40:25.242 [90019] <10.0.10.122> #3 /Applications/Server.app/Contents/ServerRoot/usr/share/devicemgr/backend/php/md m_checkin.php(140): PerformInTransaction('_checkin_transa...', Array)
0::Oct 29 13:40:25.242 [90019] <10.0.10.122> #4 /Applications/Server.app/Contents/ServerRoot/usr/share/devicemgr/backend/php/ch eckin.php(13): require_once('/Applications/S...')
0::Oct 29 13:40:25.242 [90019] <10.0.10.122> #5 {main}
1::Oct 29 13:40:25.242 [90019] <10.0.10.122> {SendFinalOutput (common.php:433)} Sent Final Output (14 bytes)
1::Oct 29 13:40:25.242 [90019] <10.0.10.122> ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ - /devicemanagement/mdm/checkin
0::Oct 29 13:40:25.242 [90019] <10.0.10.122> {SendFinalOutput (common.php:433)} Completed in 1ms | 403 Forbidden [https://server_url_obfuscated/devicemanagement/api/device/checkin]
Looks like I may have to re-enroll every device after all, unless someone has a better suggestion.