How much of the physical density of the DC area is due to Metro's presence, though? In 1950, with DC at a higher population than today and, due to the fixed boundaries, a higher density than today, the city managed just fine with streetcars (albeit ones with a few underground portions, e.g. Dupont Circle, Capitol Hill).
Metro's crush-loads could probably be accommodated just as easily on a dense network of streetcar lines as on a system of widely spaced subways. According to
this (PowerPoint warning) the number of jobs in DC has remained relatively constant since the 1950s. If we accept this as a rough proxy for
downtown DC employment, and you take into account that the majority of Metro riders are making a traditional downtown commute, I'd say the former streetcar system could easily take the role of Metro
within the district.
Of course, that also raises a whole host of other questions... assuming the streetcars still existed and Metro had not been built, would downtown DC have been able to remain the huge regional employment center that it is, or would it just be the highest levels of government rubbing shoulders with tumbleweeds and shuttered storefronts, while the millions of suburban workers, both Federal and private, commuted to suburban office parks on the (much larger) DC-area freeway system? This is why I hate hypotheticals. Metro was undoubtedly a big factor in creating the dense and dynamic capital region that exists today, but there were innumerable other factors as well, whose effect may have benefited DC even without Metro.