Being FDR is a personal name, though; an individual personality, clearly defined as such. "Captain America", as a concept (and one originally created by the government, at that), is a person trying to embody America's core ideals and live them out, while also fighting America's enemies; it just seems wrongheaded to suggest that only one guy from one period and one perspective in the broad sweep of American history could do that.
Steve was/is the exemplar, but it's an example that I think is more valuable if others can follow it.
One of my favourite lines in the whole series: "They were Captain America."