it sounds like it could be a power supply or logic board
Yes, they would be the prime suspects. An Apple Centre won't touch it as it is officially obsolete. You could try to locate an authorised Apple Repair Centre who may have the necessary spares in stock - Apple won't supply them.
It may be time to invest in an Intel based iMac!
Apple has discontinued support for certain technologically obsolete and vintage products.
Vintage products are those that were discontinued more than five and less than seven years ago. Apple has discontinued hardware service for vintage products with the following exception:
Products purchased in the state of California, United States, as required by statute. Owners of these products may obtain service and parts from Apple Service Providers within the state of California, United States.
Obsolete products are those that were discontinued more than seven years ago. Apple has discontinued all hardware service for obsolete products with no exceptions. Service providers cannot order parts for obsolete products.
All retail Apple Stores and the Canadian, European, Latin American and Asia-Pacific operating regions follow the U.S. product list, but make no distinction between vintage and obsolete. When applied to the retail Apple Stores and these operating regions, all products on the U.S. Vintage list are considered obsolete.
https://support.apple.com/en-gb/HT201624
