First, if this is your "problematic" iMac:
https://everymac.com/systems/apple/imac/specs/imac-core-i3-3.06-21-inch-aluminum-mid-2010-specs.html
then it came with an early version of Snow Leopard, OS 10.6.3, installed. Did it also come with that software on a DVD?
Also, the maximum OS that machine can run is the last version of High Sierra, OS 10.13.6. If your wife tried to install Mojave, she would have received an error message.
Next, it looks like there are no instructions for replacing the video card:
https://www.ifixit.com/Device/iMac_Intel_21.5"_EMC_2389
Note, though, that link has instructions for removing the internal drive.
Third, here are the results of a google search for "ATI Radeon HD 4670 graphics with 256 MB of dedicated GDDR3 memory":
https://www.google.com/search?q=ATI...rome..69i57.1600j0j1&sourceid=chrome&ie=UTF-8
Is there a way for you to check whether or not that machine is backed up, on your shared Apple Cloud storage? I use my own "local" drives (actually SSDs) for backups, so I really cannot help with "Cloud" features.
As for refurbished, I prefer brand new myself. Our youngest son bought as a refurbished Philips 46" Plasma TV a few years ago, but it went bad after less than 3 years of use. On the other hand, I had previously purchased a refurbished power supply "brick" for an older Mac laptop I had, and it stood up well.
How much of a budget do you have to spend on a newer iMac?