Yes, that was already clear in your initial post. And you know yourself that your laptop is defective.
The problem is that either the eDP port is not working or the internal display is broken. It looks like the HDMI port is only initialized by the booting OS, but not beforehand at the firmware level. Normally the BIOS recognizes a deactivated (eDP) port/display and sends the image output to the next active port with a connected display.
Simple example: Laptop with external monitor → switch on the laptop and close it → internal display is deactivated → port with external monitor is activated and shows everything as usual. Have you tried that?
You can take a look at this to understand what I mean: