You've posted an article that conflates the two symbols, but the caduceus means different things than the Rod of Asclepius and has different origins despite the similar appearance.
And in the age of Covid, perhaps the Caduceus "ancient and consistent associations with trade, liars, thieves, eloquence, negotiation, alchemy, and wisdom" is just perfect for the state of medical science and public health and laboratory gain of function experiments, horse dewormers, and mRNA vaccines which turn out to be no so much like the vaccines we're used to.
The point is the medical origin is older than the U.S.