I’m running DietPi on a Odroid XU4.
I’ve installed fail2ban and I should have logs but nothing.
When I look at the log file just after booting up my XU4, the init logs of fail2ban are written in the log file. But after a while, it’s simply empty.
For your purpose of use , Full (Rsyslog & Logrotate - the “standard” of Linux logging systems) can be a good choice.
Please change it via dietpi-software.
But you will loose some advantage, e.g. system performance and SDcard time to failure.