Despite the fact that you have the log posted from cron, I kinda doubt that cron is running correctly running based on my experience. Never had an issue really with cron except when cron was not set up correctly - and that a cron job is running doesn't mean that the cronjob is setup correct. Do you have access to cron directly or just via your hoster?
Btw. Usually the cronjob for osTicket should look like /usr/bin/php /path/to/osTicket/api/cron.php and not like the screenshot you posted (/usr/bin/curl ...)
I know that's may not the answer you hoped for, but that's what I think is the issue. Maybe forum moderator @[deleted] has some other / additional ideas though.
Michael