There are private donations where you can donate to family members or someone designated (these are usually living donations though, such kidneys or bone marrow, but some are post mortum) and only the parts requested/required then there is a general donation where all usable parts are harvested. The person who fills out the donation forms decides which type of donation it is and the process moves on from there.
As a parent, I can understand your position, Neph, but being on the other end of the stick really changes your perspective. I would want another kid to live and another parent to not have to worry will my kid find a heart or a lung or an eye if DC were to come to an early end. It would be hard, but I would sign the papers and let them have as much of him as they wanted.