Could a person simply put the guts of the cooler into an old fridge that has given up?
Yes, Good starting point. Make sure you understand which part goes inside and out.

4" of insulation seems like a bunch.
Yep, then again, comes down to this choice:
One time spend money on insulation or forever spend more to run it?
Make sure the extra insulation will have a payback in a reasonable time.

Check out the couple existing off grid Fridges, similar thickness to get their low numbers.

As you have seen many ideas to reduce the cost, energy, equipment etc.
Let us know what your issues are, always good to show what you have found and why you need something else. The more you share the better we can help without telling your life story!-)