Kidney transplantation, as it compares to liver transplantation, strikes me as a much lower stress affair. For starters, there are more kidneys than there are livers available for transplant (although this is somewhat negated by the fact that more people need kidneys than livers). More importantly, we can keep people alive who have no kidney function via dialysis. This affords time to search for an organ that is a really good match for the recipient. A good match minimizes immune differences (we focus on major antigens like ABO and HLA) and results in a longer lifespan of the graft (transplanted organ).
People with serious liver failure really cannot be kept alive without a transplant. The kidneys and the liver both perform a similar function: filtering toxins from the blood (among other things). Kidneys filter out substances that are ionic or polar; the liver deals with things that are not polar, or fat-soluble. Its been fairly easy for humans to develop a device that can separate ionic substances (dialysis machines) but not so easy to develop an artificial liver.
Anyhow, we can figure out in advance who will need a liver transplant and start looking for an organ for them before time runs out. The way this is accomplished is through a system (MELD) which accounts for a bunch of different factors, including some labs that assess liver and kidney function, as well as time spent waiting. The system works so that people who are closest to the end of their liver function get pushed up on the list.
One of the things that can happen with any organ transplant is a hyper-acute rejection. When this happens, the body's immune system launches an immediate attack on the new organ and the vessels end up getting clotted off and blocked. The only solution to this disaster is to remove and discard the transplanted organ. If this happens to a patient who is receiving a kidney transplant, that is bad, but at least the patient can leave and go back on dialysis. What happens if it happens to a liver transplant patient? There is no substitute.
Turns out, what happens is the patient gets moved to #1 on the national registry for livers. A new liver will be procured from somewhere in the country and delivered to the hospital within a matter of hours.