I have a bit of a unique perspective on this...
I donated a kidney to my sister. She has 3 now, I have 1! I can confirm I'm still a whole person
. It never even occurred to me to refuse to give up a kidney on ethical grounds or because I would no longer be "whole".. I guess it depends how you see the connection between body and soul.
Having a bit missing from my body hasn't changed who I am in life, so I doubt it would change who I am after death either. I'd hope my family would know me well enough to know I'd want something good to come from my death, and having seen what my sister went through on dialysis I fully support an "opt out" system. If I hadn't given her a kidney she'd have been so far down the waiting list she'd have probably still been on dialysis now, or dead. It's no way to live. I also watched a dear friend's partner go through 2 liver transplants. He's only 26. He suffers from a disorder that attacks his liver and will need another transplant in about 10 years time.
Trust me, those people who are questioning whether they'd accept an organ or not: You will take one if the time comes. You will not sit in a doctors office, yellow from jaundice or swollen with kidney failure and say "Hmm, I'm not sure about organ transplants.. I think I'll take my chances without one, thanks.. it's all just a bit weird for me!". If you're in that chair next to your child or your loved one who is dying in the bed next to you, you won't even think twice. You won't even consider the ethical implications of transplants, you'll just be trying not to die or to lose the person you love just like the rest of us.
It's so easy to say "I don't agree with organ donation" when you're standing in a healthy persons shoes, but I whole heatedly agree that if you would accept an organ for yourself or a loved one (which you would, it's not even a question) then you should be willing to donate your own organs to others.
When they finally put me in the ground after a (hopefully!) long and happy life, it will be my inside self that people will miss and think about, not my body or my organs so I would never begrudge my lovely sister for taking a piece of me, and when I'm gone I hope the rest of me is taken for whoever needs it too. I hope the rest of my organs give someone else the chance to live long and happy, and I'd hope my loved ones would find comfort in that :)