Bodies are a kind of use it or lose it deal. If you are buried, everything except your bones and teeth will be eaten. Organ donation is the least of your sovereignty problems.
For instance, send a letter every 6 months where you tell of the social benefits of donating your organs and thank-you notes from receivers. Sponsor national organ donation days. Even properly aligned incentives: Make certain government-sponsored health care cheaper and more available for donors than non-donors.
But make any opt-in cryptographically verifiable, and signed with an e-ID. I really don't want my opt-out hacked during DEFCON 2023. And implement it correctly: often government attempts to curb self-ownership are well-intented (physical and mental healthcare), but ill-executed (block internet porn, close fast food restaurants, ban psychedelic drugs) and they compound.
I full well realize that anything that happens after my death is out of my hands. Even if I don't donate now and opt-out, maybe in a 1000 years they can clone some of my DNA to grow new organs. It is purely about opt-in vs. opt-out for me (and about democratically consulting the majority opinion on an opt-out).
Opt-out (happens during life and on life-support) of self-ownership [1] is, to me, a violation of human rights. One of the most fundamental and innate rights, which should be a given (but could be opted out of by opting-in for donorship). To partly take away a right, unless you say "no!" sets a dangerous precedent.
> Everyone's right to life shall be protected by law. No-one shall be deprived of his life intentionally save in the execution of a sentence of a court following his conviction of a crime for which this penalty is provided by law. - Human Rights Act 1998
> The legitimate powers of government extend to such acts only as are injurious to others. [not: are beneficial to others, unless you want to posit that opting out of donorship injures others in need] - Jefferson
So it boils down to what do you consider death?
> My impression is this is only true for some transplant types.
Other way around. Organ donation after cardiac death is only possible for certain types, and only in very limited circumstances.
Hopefully things will improve because it would vastly increase the number of possible donations. The vast majority of people can not donate organs even if they want to - it requires a very specific type of death (brain death where the body is just fine).
Many people also have religious and ethical problems with it because you are killing someone. Obviously that depends on how they define death, which has no commonly accepted definition, the current definition "brain death" was set that way to make organ donation possible, and that makes many people uneasy.