That's not how it works, and it's sad that media and other outlets perpetuate these stereotypes.
Here's what happens if you're in a serious accident (to both you and your wallet):
The second you arrive in the ED triage bay, we start cutting off all your clothes. We do this to expose any hidden injuries and assess you fully and quickly. 100% of our focus and effort is on trying to ensure your safety. We won't even know your name at that point, unless you are awake enough to tell it to us. We treat everyone the exact same.
Your clothes and belongings will be put in a bag and given to hospital security. If you have an ID on you, the social worker on call will use that to identify you and start working on figuring out how to contact your family. These social workers, by the way, are miracle workers at finding and contacting people quickly. I have no clue how they even accomplish it half the time. If you are an organ donor, that would get noted in your medical record.
Meanwhile, we are still working on you. No thought, consideration, or pause is given to organ donation status by the providers caring for you.
The ONLY time we handle a patient differently is when we receive clear and immediate information that a patient is, say, on comfort care or DNAR/DNI.
Organ donation status has zero impact on the quality of care you receive or on the intent of treatment. We literally don't even think about it.