This actually is not accurate. There were more bureaucrats in Richmond than DC, and Richmond was a vitally important industrial center and one of the only places the South could manufacture their own artillery and weapons in any quantity. Further, without Richmond, there is no effectively defensible point in Virginia, and the whole state is soon in Union hands. Without Virginia's wealth, resources, and manpower, it's hard to see Richmond's fall as not having a major impact.
And you're right about the industry in Richmond. Where are your figures for the number of bureaucrats coming from? And anyway, it's less about the number of bureaucrats since they could always move to another city. It's the records and other governmental infrastructure that would be lost if the North lost Washington. I still can't believe the Confederacy had much governmental infrastructure in April '61. The North having an established government was one of their advantages over the South because the North didn't have to build a government from scratch. Since the South was working on a more ad hoc basis, it would have been easier to shift the Confederate government to another city.