Fixing My System76 Laptop in 20 Minutes or Less

Minutes before a recent video call where I was slated to give a demo, I initiated a reboot on my System76 Oryx Pro laptop, but it did not come back up after seemingly powering off. I attempted to manually power on and found it unresponsive. The machine was connected to power, so I saw the familiar power and battery LEDs illuminated on the front of the machine.
Given that the machine appeared to not even be attempting to boot, I suspected that the issue was likely related to power or the battery.