A cure-cancer PoW where each block fixes all of cancer is ridiculous.
A more apt analogy would be that each block would solve a protein folding or active site matching problem for cancer. Then each solution would be public, and useless to mine over again.
Except the miners don't gain value from discovering a protein fit, so they have no incentive to mine the wrong chain.
Even then, as long as the coin value is larger you will have a consensus with the miners to to mine the latest block. Any sidechains will be neglected, which is why this "no value or nothing" attitude doesn't fit a blockchain network.
A more apt analogy would be that each block would solve a protein folding or active site matching problem for cancer. Then each solution would be public, and useless to mine over again.