This is such an interesting thread but I want to add a small but important point which seems to have been overlooked. It is Not the Case that people with serious illnesses can never, themselves, become organ donors. They can often still donate tissues - skin, bone, tendons, corneas, heart valves (sorry if TMI).
So, it is worth signing up to the Organ Donor Register, even if you think you're poor old body is too sick to be of any use to anyone. There are very few illnesses that prevent donation, and the medical staff concerned take a full medical history to check a potential donor's suitability.
I'm on the Register, and, quite frankly, am considering leaving my body to science (perhaps with appropriate tattoos - "here's the liver - one not very careful owner") .
The body is just an envelope after all. And, in case anyone is interested, I spoke to a medical student recently and she said every cadavar they use for dissection is given a proper burial, which the family concerned are invited to attend. I don't mean to upset anyone by being so frank and apologise if TMI but I am thinking of the three people who die every day because a suitable organ can't be found.