This is an understandable misconception but also one that medicine corrects for. Firstly, the team taking care of a patient never knows a patient's organ donor

This is an understandable misconception but also one that medicine corrects for. Firstly, the team taking care of a patient never knows a patient's organ donor status. Only a few people have access to the status and they are called in to get the organs only after the care team has declared the patient dead. Even the numbers are usually in favor of the patient not being a donor given not many people register. So the most likely outcome of letting a patient die is just death not organ donation. The team treating the patient is also made of human doctors and nurses who really don't like to kill people, their entire job is the opposite of that. Their drive to save the patient in front of them is higher than saving an abstract person somewhere else who may not even exist because organs last less than 24 hours outside of the body.