Actually, this does pass the (rotten) smell test.
Mammals, which includes humans and cows, can metabolize diclofenac. Vultures cannot - hence it is toxic to them but (at reasonable doses), not toxic to us.
After reading the research I think there is some validity to the hypothesis. In the US we can't use those NSAIDS in cattle because of residue. Europe, India and Africa are not as stringent about drug use in food animals as the FDA.