I can promise that this one will be as disinterested and objective as possible, as I labour under quite a bad cold, and so have every reason to wish for the indicated cure. Whether it will be amusing or make sense is obviously another matter.
The traditional appositional phrase is “They can put a man on the Moon” but this is now an obvious archaism. It is revealing, however, that the preferred technological marvel is one which was signally useless, albeit highly remarkable. If we replace it with various other prodigies of science –
They can transport me to the other side of the planet in twenty hours instead of twenty weeks, but they can’t find a cure for the common cold.
They can detect and treat a wide range of illnesses, almost all of which were fatal less than a century ago, but they can’t find a cure for the common cold.
They can repair and rehabilitate a person whose spine was severed but they can’t find a cure for the common cold.
– the relative insigificance of colds will readily become apparent.
Once the common cold is cured, we can anticipate a future variant of this phrase which seeks out the next most nugatory of life’s discomforts:
They can put a man on the Moon, but they can’t stop me from forgetting where I put my glasses.
They can put a man on the Moon, but they can’t prevent my father-in-law telling the same jokes over and over again.