Argon One Always On Not Functioning

Are you sure, that the system was shutdown properly before? Normally the system should start with a button press again, also if the jumper is set to the always-on position. Or do you everytime unplug the power supply from the socket? Can you verify that, please?

Depending on the operating system you use, you should check if the system is booted headless and you can login via SSH. If the system is booted, but only no video output, then it could be an issue with EDID data and you could force the video resolution via kernel command line.