Why does my iMac shut off halfway through boot up

There are a few different issues you could be facing. But you will need to take the system apart and using a DVM test some voltages. If you haven’t worked on a TV or computer at this level it might be wise to find another shop that knows up from down!

Mac’s of this vintage uses CCFL’s, they use high voltage and the power supply uses capacitors which hold quite a wallop if you touch them. So be careful!!

I would start off checking the display using a flashlight pressed on the screen you should see the faint image of your desktop and its icons once the display goes out. If yo do then you know the displays backlight has problems which can be the inverter or the CCFL tubes are worn out.

The next is the power supply following this guide iMac Intel 20" EMC 2133 and 2210 Power Supply Replacement as this system could have bad capacitors Capacitor Plague

Then either getting a new one or get the needed replacement capacitors.

If it’s OK I would then check the logic board power rails but this gets difficult if you don’t have the schematics and boardview drawings to referece.