This is a situation where you definitely can't say beforehand what you would do. You just don't know how you will deal with the grief and the pressure at that time.
Waiting for my dad to die, I barely slept (the hospital advised we try but didn't really have a big enough relatives room to rent for us all, plus, my dad was dying!!
), I barely ate, I had a dodgy tummy for the full 12 hours (stress screws my digestive system), and I had cried so much I physically ached. I actually needed to cry at a few points but it hurt to do so and so I couldn't.
I'm a sensible, intelligent, generous, non-religious (I don't believe you need your body complete for heaven or anything) person. I understand the benefits of organ donation. However, after 12 hours of suffering from that initial raw, smack in the face of grief, if they had mentioned donating his organs, I can't tell you what I would have said. I like to think I would have said yes, after all, what is another few hours of my grief if another family doesn't have to grieve? But I can't guarantee that.
Fortunately my situation is hypothetical, I didn't have that question. But I can see exactly why people might say no.
To not understand this is to be a bit ignorant IMO.