They did not claim that AI will cure diseases. Only that it would speed up the process, potentially help curing some diseases. That’s a very different claim.

"I think it will actually be possible to cure most human disease in ~5-10 years, as crazy as it may sound" is a direct quote from Dario Amodei. In his original essay he makes clear that that doesn't mean all those cures will be distributed to all on that time-scale, but I would still interpret this as a wildly ambitious prediction that AI will cure most diseases. He predicts in the same essay that it will be responsible for a 10X speedup in drug development.

Again, they did not claim AI would do it. Only that it would help speeding up biology and medicine.

> The bulk of the essay is devoted to refuting skepticism of AI’s potential in health and biology, and showing why I think it will actually be possible to cure most human disease in ~5-10 years, as crazy as it may sound to ordinary people and frankly to biologists as well (I used to be one!).

https://x.com/DarioAmodei/status/2088758816376807762

The paraphrased version seems accurate to me, based on this tweet which I assume is the source.