Non-removeable graphics hardware started in 2012 iMacs
Those models have socketed graphics and the 6970 was a popular after-market upgrade swap in both 2010 and 2011 iMacs. This is the card (lower right) and its complicated heat sink from a 2011 iMac. The card is MXM form factor, but no MXM cards other than those made for Apple had the required Mac ROMs.
The desirable 6970M was always in short supply and not completely trouble-free, yet the card today costs far more than the computer is worth should you even manage to find one in usable condition. Even replacement 5750s are pricey and some on the used/pulled markets have already failed. That is why they were "pulled."
Installation is not for the faint of heart.
I had to say goodbye to my 2010 27-inch when its 5750 card died because, even though I have some skills with working inside Macs, I could not find a replacement card that stood up to a simple cost/benefit test for a long-obsolete Mac. It would have been cheaper to buy a used, working 2010 iMac 27. I bought new instead and never had a reason to look back.
Like the others, I cannot recommend trying to replace the card, which may not the be the only issue inside a broken 11-year old Mac.