Ah yes, the old symbol of trade, economy, and messaging... a real medical symbol (the one with only one worm) would make more sense
Oh, don’t even go there... (!).

I fully agree — but the stylized Starfleet version of the caduceus is at least a well-established symbol in-universe...
Whether the use of the caduceus by the US Army was a mistake, and a complete misunderstanding of classical symbology, or whether it really was intended simply to signify medical staff as “non-combatant,” as some have argued, I doubt we will ever truly know.
Either way, the single serpent on the staff of Asklepios, Greek god of healing, would be the more correct classical symbol for medicine. It would sure fit into the centre of the delta badges more easily!
Fletcher’s version of the caduceus is clever because it moves on from the Greek original:
The simplified coils of the entwined twin serpents from the classical caduceus now suggest the double-helix structure of DNA — which is a really neat idea and is a pretty universal concept for organic life throughout Trek. (Yes, OK, silicon-based Horta and whatever the Tholians are...) I guess the staff and wings now becomes almost an angelic figure — maybe it can be argued to have some significance to other Federation species beyond just the human influence...?
It’s stretching a point — but at least by
not using a traditional caduceus, it leaves the door open for it to be something other than a historical mistake, even though it’s also a symbol still recognizable to the contemporary audience.