I want it to time out after 5 or 10 minutes of no login and shutdown.

Maybe grub timeout feature satisfy you.

Add custom menuentry with halt command into the grub configuration and make it default. Set grub timeout to needed value. After poweron or reboot grub selects your default entry and turn off the computer after timeout if there is no user activity,

To boot any OS you need to select and boot menu entry manually.

I want it to time out after 5 or 10 minutes of no login and shutdown.

Maybe grub timeout feature satisfy you.

Add custom menuentry with halt command into the grub configuration and make it default. Set grub timeout to needed value. After poweron or reboot grub selects your default entry and turn off the computer after timeout if there is no user activity,

To boot any OS you need to select and boot menu entry manually.

For Arch add into /etc/grub.d/40_custom

menuentry "System shutdown" { echo "System shutting down..." halt } 

To change the default selected entry, edit /etc/default/grub and change the value of GRUB_DEFAULT:

GRUB_DEFAULT='System shutdown' 

then

sudo grub-mkconfig -o /boot/grub/grub.cfg