"Brain death" is an unreliable indicator of death. A similar analogy would be someone who has stopped breathing. Yes, breathing may have ceased because that one is dead - dead people don't breathe, either. To write someone off as dead just because he has stopped breathing will usually end up in real death. However, by taking prompt action when someone stops breathing, a life can often be saved. The same applies for cases of "brain death". There are those who have been diagnosed as brain dead who have later recalled the debates which took place at their bedsides about turning off their life support.
If brain dead people were really dead, they wouldn't respond as they do to organ harvesting. The below is only a news article, but I think it describes my point - if one can feel pain and needs anesthetic for surgery, one is not dead.
"The concerns come from the fact that if a patient is not sedated during procedures to remove heart, lung, liver and pancreas, there is often an alarming and dramatic response from the body."
"Anaesthetists have observed that patients' pulse and blood pressure shoot up when the first cut is made. Theatre staff are often distressed when clinically dead patients move and wriggle about, to the point where it is impossible to operate."
"The editorial claimed that studies showed there was some level of activity in brain cells, even when the brain stem (which connects the brain to the spine) was not active. However, the guidelines said that dead patients do not require analgesia or sedation.
It added that dead people did not require anaesthesia, and if a person was not dead, they should not be having their organs taken away."
https://www.abc.net.au/science/articles/2000/08/22/166001.htm