Mid-2011 27" iMac random reboots, GPU defective?

As the neighborhood computer repair guy, my friend gave me a broken iMac and told me that if I could fix it I could keep it. The exact details are:

iMac mid-2011 27"

3.1GHz i5

1TB HDD

8GB RAM

AMD Radeon HD 6970m 2GB graphics card

It is randomly rebooting, often just seconds after power up (e.g. it can't make it to the login screen). Its as if someone completely pulled the plug on it and then the computer tries to turn back on.

When the issue first came up, he brought to to Apple and they replaced the PSU. The problem still happened. Then they replaced the logic board. No luck. They gave up at this point, and he gave it to me.

The first thing I thought, if it wasn't the PSU or logic board, is that I should try just simply swapping the RAM. No luck. Then I tried replacing the HDD with an SSD. The computer made it through the OSX install just fine, and when it boots its wicked fast. I thought I had fixed it. Then the next morning I got up, powered it up, and it very rarely makes it to the login screen - the problem was still there.

At this point, every single part in this computer has been replaced and/or tested except the graphics card (Radeon HD 6970m 2GB) and the optical drive. I am also aware of a defective line of these exact iMacs with the exact same graphics card that Apple actually issued a recall on (see here: https://support.apple.com/en-us/HT203787) Unfortunately this particular iMac was purchased more than 4 years ago so Apple won't replace it.

Is my problem most likely the video card? Am I stupid to go and spend $250 on another one off eBay, or is this computer simply cursed.

Thanks for any help. I would really like to get this working - its a very nice computer and as an amateur video editor it would be wonderful to have a decently powerful Mac (also 4K displays are nice!).

TL;DR

Got a 27" iMac free. It randomly reboots. Have tried replacing every component except graphics card. Can a graphics card cause random reboots, and is it worth replacing?

You may find this question of interest to you.

Unfortunately, this iMac still has intermittent problems and sometimes boots and sometimes doesn't. When it doesn't it just keeps restarting, and even when it does boot it will randomly reboot at some point.

However, to answer your question, yes, a graphics card can cause random reboots. Whether or not it's worth $250 to replace only you can answer. Obviously, if it fixes the problem it's money well spent. If it doesn't, then not.

But it's what I wanted to do next on the iMac I was troubleshooting, but the owner decided to call it quits. If it was me, I'd personally take the plunge and if it didn't resolve the problem I'd try and resell the graphics card. But that's a personal choice on my behalf and you need to do what you're comfortable doing.