If your site is up at 2.2.9+, remove the search block from the front page and remove the search block altogether. Not only delete within the Moodle interface, but maually remove a 'search' folder from moodlecode/blocks/.
The try your move to 2.3.6+.
The missing from disk, in the past, wouldn't prevent one from going forward with the upgrade. Although, if one didn't delete it from within Moodle, every time one did a plugins check, Moodle would complain.
On your move from 2.2.x+ to the 2.3.6+ were permissions/ownerships on all files and folders set to the web server user? Error 500's could be many things.
One could use the config-dist.php file and copy the debugging lines from it to the config.php file to see if Moodle can report anything at all.
'spirit of sharing', Ken