Coupla more things:
Yes, there is a rapid HIV test available now, but if it comes up positive they still have to do the Western Blot test to confirm. I have no idea what the incidence of false positives is on rapid-HIVs but it’s probably pretty low.
And (AFAIK - I’m not a nurse but I’ve been involved in a couple of harvests) they have to run tissue-typing tests to see who the organs can go into. There’s pretty much no way to get an organ out of the donor and into the recipient in 30 minutes or less (I’m exaggerating). That’s why the person has to be on life support.