I am an organ donor, but I admit to being squeamish about going whole-hog and donating my entire body.
Anyhow - it's pretty simple. I'm done with it, so have at it. Even after decades of self-abuse, there's probably something or other that could help someone else. It's surprising what they need. We tend to focus on the big bits - heart, liver, penis (hey, for someone with no penis, mine would be an upgrade) but there are plenty of weird little things they also need.
If people are squeamish about everything, I don't really have a problem with it. I would never promote a tit-for-tat system (if you don't agree to donate, you're off recipient lists). I do support opt-out, though, or at least making giving an answer mandatory and binding (see below). One of the near-universal truths of humanity is that we are lazy as all get-out, unless motivated by an external force. Imagine a country where everyone would have to automatically serve two years in the army unless they opted-out.
Attaching the organ donation question to a drivers license is an attempt at making it universal (and a bit morbid, when you think about it - "Here's your motorcycle license, and you really, like really-really, should fill-out this card"). Perhaps it could be a part of something that is renewed more often, and even more universal. Let's say in Canada, you don't get a health card unless a response is provided. Income tax filings might also work (as inevitable as death, as the saying goes).
Where I live, it also depends on the family. When one is near death, there are various conversations between the medical people and the family. One is about organ donation. It really doesn't matter what the patient wanted, though if it's a "yes", that can help guide the family. Families may also have heard the wishes of the patient before (I have communicated mine), but feel differently when it's crunch time. Even the most gung-ho family may hesitate when it comes to children - and yes, children need transplants.
Finally, a couple random thoughts. If you don't wish to donate, but you do wish to be cremated, I think you're a bit weird. I can't express exactly why, but in my fickle mind, it doesn't make sense. If you want to be like Tutankhamun or Ramses and show the world your smile long after your death, know that the nice people who mummified them emptied them - including sucking the brain through the nose - to get that certain je ne sais quoi of not rotting. If you want to really do it right, you're going to have to find the right bog or glacier to work with.