I know it is not really what is asked, but cron is a pain in the ass to handle and manage. I am not sure if it is officially deprecated yet, but I would migrate everything to systemd timers instead it is so much better. It provides configuration tools and proper integrated logging and troubleshooting tools.
Just create a service file of type oneshot which runs your backup script and a timer unit with the same base name. Set the timer to hourly, place both files into /etc/systemd/system, do a daemon-reload and enable the timer. You can see the status or journal for output and list-timers to see the schedule and wether or not it ran.
Usually if programs can run in a user context but don’t work as some automated process it is either due to environment differences. Most importantly PATH which can be solved by using absolute paths for programs. Another very common problem is the systems MAC implantation although it happens more often with SEL. Still you might want to check your AppArmor configuration and (audit) logs.
If you want to stick with cron also make sure to read the mails (/var/mail/root by default), because most cron implementations dump their output/logs there.