iMac shut down during loading, then the screen remains black

I think you’ll need to open the system up so you can see the onboard diagnostic LED’s. Follow this guide to take the display off iMac Intel 21.5" EMC 2428 Display Replacement just be careful you don’t tug on the cables are its very easy to damage the logic board connectors

With the display off locate the four LED’s in the middle part of the logic board

Here's a breakdown of the onboard diagnostic LED's:

LED 1 - Indicates that the trickle voltage from the power supply is detected by the main logic board. This LED will remain ON while the iMac is connected to the AC power. The LED will remain on even when the computer has been shut down or put to sleep. The LED will turn off only if the AC power is disconnected or the power supply is faulty.

LED 2 - Indicates that the main logic board has detected proper power from the power supply when the computer is turned on. This LED will be ON when the computer is turned on and the power supply is working correctly.

LED 3 - Indicates that the computer and the video card are communicating. This LED will be ON when the computer is communicating properly with the video card. If LEDs 1 and 2 are ON and you heard the startup sound, but LED 3 is OFF, then the video card might be installed incorrectly or need replacement.

LED 4 - Indicates that the computer and the LCD display panel are communicating. This LED will be ON when the computer is turned on and video signal is being generated. If the LED is ON and there is no image on the LCD display panel, the LCD display panel or inverter might be installed incorrectly or need replacement.

Remember LED 4 won’t light as the display is disconnected.

Let us know what you discover.