Don't worry, I'm sure that's his next blog post. Right after Claude has finished solving famine and poverty. ;)
It is possible to prevent most viruses by recoding the human genome (requires embryo engineering or pre-embryo engineering) to use alternate codons that existing natty viruses don't use. This wouldn't prevent synthetic viruses from being developed that use the alternate codons, of course. See (1) https://diyhpl.us/wiki/transcripts/hgp-write/2016-05-10/ultr... and (2) https://diyhpl.us/wiki/transcripts/hgp-write/2017-05-09/ultr... and (3) https://x.com/RokoMijic/status/1720875112541364535 for some background.
As for "curing cancer", I suspect that Anthropic or OpenAI will use their high status/reputation to launder the existing solutions (germline, p53, enhanced DNA damage repair pathways, higher tumor-suppressor-to-oncogene copy number, more robust tumor-suppressor networks, more self-anti-cancer apoptosis, CD47 suppression, etc) that society has refused to accept. They might have enough reputation to get people to look at these ideas. It won't cure people with stage 5 turbo-cancer a.k.a the walking dead, but hopefully we won't be moving the goal posts _that_ much on them.
A more sophisticated view is that cancer (one idea) is an problem of rates (like most things in biology) and there is a tumor burden per unit time from whatever causes vs. the immune system's capacity to detect and destroy them per unit time. As humans age, the balance tips in one direction. If the balance is too far off for too long, tumors accumulate, doctors take notice, and assign whatever labels to the condition.
Looking into particular causes of tumors (and calling each cause its own cancer) can create a vast research program that employs a lot of people, since there are so many mutations that lead to tumors. But interrupting the development of a particular mutation is not going to have the same society-changing effect that solving the fundamental issue of rates would.
Literally first phrase on Cancer Wikipedia entry.... [1]
"Cancer is a group of diseases involving uncontrolled cell growth typically resulting in tumors with the potential to invade or spread to other parts of the body...."