I hope that one day you have no choice about whether your organs stay in your dead body or not.
Once you are dead, you are property, not a person. The person has gone, depending on your beliefs, to the afterlife or to oblivion.
Keeping valuable, life-saving organs inside a dead body because of your beliefs is morally wrong.
Who's property? The gov't? Some big proxy business contracted by the gov't? The family? If family? Who gets priority?
Will it be donated? Or sold? If its gonna be donated and property of family... It will still be the same as today, unless the way people think has changed drastically in that future of yours.
If its owned by gov't... nationalism will have to be pretty damned strong... for it to even happen. And there will be people fighting against it, no matter what, every step of the way. There will be people, who would destroy their own organ and body, upon their deathbed or commit suicide when they know they are going to die anyways, to ensure noone gets their grubby hands on it and/or to get the point across.
It'll be a damn hotbed of a political maze with traps scattered about like abortion... Probably worse... since it would effect everyone. (At least from a happening in America viewpoint...)
Not a lot of people want to know that their body will be screwed around with after death, not a lot of family want to see/know their dead relatives become property of someone else and screwed around with. Burials, cremations and all that fancy stuff exist because of that. Respecting the dead and all that.
Your vision is... can't say I like the idea of being someone's property... I can imagine being put in a morgue, but more akin to a butcher's freezer for practicality and expense purposes then a standard morgue... If its proxy business run, the bodies would probably just be stacked and/or not properly body-bagged if they can get away with it. They'd have showcase freezers and the freezers where all the bodies normally stay...
I predict that it is unlikely to happen... until after humanity's survival factor kicks in... and dead humans have some way to fix it...