Why wouldn't you want to donate your organs though? And especially: Why wouldn't the family want to donate the organs of their relative?

Families are typically out of their depth in these situations and completely dependent on medical staff to guide them in decisions regarding their loved one. Medical staff may confront the family with the question of organ donation before they have been able to internalize the decision to withdrawl life support. Without time to internalize the decision to withdraw life support families may feel that allowing organ donation may be bias medical staff and, even subconsciously, tip the scales.

Families may be uncomfortable with yielding control of their loved ones body to the medical establishment. The question of organ donation is presented as a binary choice to families (all or nothing) and requires a significant degree of trust in the medical community. They may worry that they are unwittingly consenting to donations of germ cells; that part of their loved one may end up in a student lab or that organ donation may delay a funeral.

Families may worry that donation will incur additional costs or that someone will profit from the deceased organs (see Henrietta Lacks and HeLa).

Also making an opt-in decision in any complex situation that requires unanimous consent of what is essentially a committee is typically going to result in no-action.

Some people are afraid that doctors will make less of an effort to keep you alive if you're an organ donor.

Knowing doctors I find that fear extremely implausible, but it exists.

That's a different issue. Some people are religious or superstitious, and I guess there are religions against organ donation.

Why can't we expect people to have reasonable reasons for their refusal?

Reasonable reasons unrelated to superstition? Name one?

I can't think of any, but I do strongly believe that anyone who can come up with one should be allowed to keep their organs :)

This would render the whole point moot. Why have a default in the first place if you always require an affirmative choice, either from the person or from the family?

That is basically the practical problem with an opt out system. Defaults are sufficiently powerful that lack of an opt out can't really (and really shouldn't) be taken as an affirmative choice.

Well, ideally yes you'd just enforce an affirmative choice here, but I think the whole point of having an organ donor registry is for time-critical situations like when you can't find someone to ask or when there is not clear person to ask.