It's a reach to imagine your scenario applies to snakebite emergencies generally. Copperheads are large, distinctive snakes, their nomenclature is stable and in the temperate zones where they live, there are a relatively limited number of other venomous species to distinguish, hence you being so assured of supplying remains for a nurse to casually ID. In contrast, many tropical habitats, where snakebites are much more of a problem, have a much, much higher venomous snake species diversity (particularly of those small, fast elusive ones you barely glimpse), and questions about ID and species relationships are correspondingly more confusing.
It's not so much about always turning up with a scientific name. The whole effort from doctor-patient communication, to animal control, to the production, supply and procurement of specific antivenins, needs to be conducted within as stable and precise nomenclatural framework as possible.
Snakes are just one example here, there are many examples of research programmes being derailed by faulty taxonomy, with big impacts on everything from crop production to theoretical work in ecology.