What a mess.
It's amazing how working with someone you know and love makes a difference.
It sounds silly but maybe it can be an add-on to a social-sharing app as preference. Something like "notify friends, friends of friends or public anonymously you are an organ donor".
Why not eliminate the hurdle of finding a match within your circle of contacts and make it an option to anonymously share that you are looking for an organ / are an organ donor?