They are brain stem dead yes. But their heart is beating. That's how donation of major organs works. They are not legally dead.
They are alive in ITU prior to going to theatre. Their heart is beating, their skin is warm, they have colour.
They die in theatre when their organs are removed and placed in ice to be couriered to the waiting recipients.
It is an enormously complicated procedure and process. The number of possible donors will always be small because the way in which you die is unusual- it will often be from an accident or sometimes a spontaneous brain haemorrhage.
Post death donation is limited to some tissues, and bone I think. But major organs require a blood supply and very minimal time without circulation. For this reason the donor cannot be dead before the recipient is already in theatre and the cavity is ready to receive the new organ.
Describing anyone who has reticence or unease about any of this as selfish is unbelievably wrong.
Families of donors go through huge amounts to enable donation. They should be praised. But those who can't should not be in anyway criticised.