Well, if your root filesystem is on an SD card, maybe you can just remove it and read files in /var/log from another machine ?
Or boot to runlevel 3 (if that Debian is using System V init) by adding "3" to the kernel command line (in the boot loader, if you can do that). Or if that Debian uses systemd (more likely) you might add "systemd.unit=multi-user.target"
You can maybe add "single" to the kernel command line to bott with fewer services started.
It looks like your display manager fails to start, then it is automatically restarted, fails again... forever.
Sorry I don't know how the Pyra OS is setup exactly. Not sure if it helps.
If you can manage to login as root, maybe you have time to type
systemctl stop display-manager
or
/etc/init.d/lightdm stop
I'm not sure.