IMac – What programs actually use the memory on a Graphics card

Recycle or sell it for parts. Since the GPU is soldered, you likely cannot bypass the temperature sensors that the system monitors to ensure safe operation with consumer skills at soldering. Have you looked into a logic board repair or buying a Mac with a shattered screen at low cost to assemble one workable Mac?

Operating a machine where you know the protection mechanisms are shutting it down to prevent overheating and persist in doing so, being frugal crosses a line into taking risks with fire if you don't go to great lengths to protect yourself and other property nearby.

Also - your characterization of design flaws seems unsupported with one link to a 540 entry discussion thread. A six year old computer having failures seems more like a car needing an engine rebuild after things start failing and not so much a design failure, but perhaps the link wasn't correct.

The 2010 11 inch Air is so much faster than a Core 2 Duo - top of the line, 15 inch MacBook pro that it's really not much of a comparison between the newer 2013 Air and anything that isn't a pro based logic board (the 27 inch iMacs tend to be the highest of end "mobile" processors with logic boards that tend towards the pro range).

Unless you need a larger display, I see no reason to shift your entire workload to any of the three last Air machines if you don't wait undue minutes for builds in Xcode to complete. From where I sit, the SSD drives and the 2013 processor lineup make for extremely productive workstations in the Air form factor.