that would kill almost anything else

Guess what: It's fact check time!!

Anthrax: ❌

Most birds are naturally resistant to anthrax because of their higher body temperature.

Botulism (actually botox, botulism is the illness caused by botox. You can't quite eat an illness): ✅

"Botulism is one of the most toxic substances in the world, and [turkey vultures'] guts are filled with this bacteria, which apparently has no effect on them," says Graves.

(Yes, the article made the same mistake)

Rabies: ❌

All warm-blooded species, including humans, may become infected with the rabies virus. Birds were first artificially infected with rabies in 1884; however, infected birds are largely, if not wholly, asymptomatic, and recover.

Cholera (actually avian cholera, which is distinct from "normal" cholera): ❓

Besides chickens, the disease also concerns turkeys, ducks, geese, raptors, and canaries.

So 2/4 diseases are already irrelevant for birds but they sound scary because they infect humans. But:

Avian cholera barely even counts because it's caused by "Pasteurella multocida" while (human) cholera is caused by "Vibrio cholerae". The two bacteria are genetically as closely related as humans and mammoths. And avian cholera does not kill anything but birds.

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This isn't to degrade the vulture's ability to withstand pathogens and toxins, that is still very impressive. It's just that obviously birds are immune to most human diseases, as pathogens don't tend to easily jump between distant species.

And even if they DO jump between species, they usually do not bring their "deadliness" with them (unless they produce poison like botox) as different species react completely differently. Maybe rabies is the one exception, as adjusting the behavior of your host by rummaging around in the brain has some universally negative side effects.