So, it was the GPU on the Logic board.
Well, if this is soldered to the board, in the 24 inch screen models, this explains why Apple just wants to switch out the Logic board.
Obviously, the Apple Stores have no real competent repair people who know how to work on logic boards or know how to desolder or solder old and new conponents off of the board.
I hope this is a cheaper repair alternative for you and I am glad you able to find a solution that works for you.
I hope they get your iMac back to good running condition.
That said, I don't like the repair service of the Apple Stores for this very reason.
Instead of identifying and fixing the issue, because of the limited repair facilities, at these stores, they just want to replace whole assembled conponents that make the repair much more costly.
I am glad and really grateful to have another, more competent Apple reseller and repair center that is not very far from where I live (about 20-25 miles or so). They have much better skilled technicians there than the ones Apple hires for their stores.
I had them do my Seagate 1TB hard drive replacement and had them do a complete diagnosis of my iMac for any other issues as I still had about a week of extended AppleCare left on my iMac.
I am happy that you finally got this resolved.
Hope you are feeling better about this, now.
Good luck to you!