Here is a timeline of developments in the space over time, created by Mati Roy: https://timelines.issarice.com/wiki/Timeline_of_brain_preser...
When we are using cryopreserved organs, then preserving the brain becomes more obviously pragmatic.
I am guessing that cryopreserved human kidneys do not have uniform cooling, spoiling interior tissue. A rabbit kidney is much smaller, and wouldn't suffer as much from this problem.
The interior veins and arteries of a kidney look like it would be a reasonably good heat exchanger.