Change the xbacklight -inc $brightness_step -time 5 in the case statement to brightnessctl -c backlight set +5%.
xbacklight -inc $brightness_step -time 5
case
brightnessctl -c backlight set +5%
Change the xbacklight -dec $brightness_step -time 5 in the case statement to brightnessctl -c backlight set 5%-.
xbacklight -dec $brightness_step -time 5
brightnessctl -c backlight set 5%-