Edit: If you use a sd card as OS drive, making a disk image should be a good option, some of the steps in the linked post should be the same, the tech might be different. I use the gnome disks app to create backup images of running pi systems on SD cards that would take longer then a manual config to restore.
Have you tried timeshift-launcher from terminal? If there are errors it will print in the terminal.
It seems a bit the application is not correctly reported to polkit.
Please click on Podrobnosti to see which application requests access.
#!/bin/bash
app_command='timeshift-gtk'
if [ "$(id -u)" -eq 0 ]; then
# user is admin
${app_command}
else
# user is not admin
if echo $- | grep "i" >/dev/null 2>&1; then
# script is running in interactive mode
su - -c "${app_command}"
else
# script is running in non-interactive mode
if [ "$XDG_SESSION_TYPE" = "wayland" ] ; then
xhost +SI:localuser:root
pkexec ${app_command}
xhost -SI:localuser:root
xhost
elif command -v pkexec >/dev/null 2>&1; then
pkexec ${app_command}
elif command -v sudo >/dev/null 2>&1; then
x-terminal-emulator -e "sudo ${app_command}"
elif command -v su >/dev/null 2>&1; then
x-terminal-emulator -e "su - -c '${app_command}'"
else
x-terminal-emulator -e "echo 'Command must be run as root user: ${app_command}'"
fi
fi
fi
As @raguse already pointed out, it should be timeshift-launcher. This is also how it is by default, so you had to change it to sudo timeshift-gtk yourself.