You know, I hear those stories - that hospitals won't work as hard for you if they know you are a donor. Yet I've not seen any numbers on this. The same folks usually state things about an open casket and other such things. I'm guessing it happens, but I'm also guessing that most times, this isn't on the Doctor's and Nurses's minds in an emergency. Besides that, one can have some policies that help negate this: For example, no financial kickbacks.

I suspect that if organ donation is widespread, it becomes much more normal, and this would cease to be an issue/rumor at all. And I do think the OP is correct - the bigger problem is families and religious views more than folks simply letting people die.