I don't think that's accurate. While many recommended such a step, and there would have been little to stop VA troops from taking DC in the first few weeks, no such attempt was made.
First Bull Run was fought when Union forces advanced into VA, heading for Richmond, and Confederate forces resisted. It is notable that CSA forces did not pursue or attempt to capture DC after their victory, though that may have been because they were about as disorganized as the Union army was.
Once cut off, the goal was to either capture the Confederates (a tall order) or force them south of the Rappahannock, since the Union military's consensus was that this river would serve as a much more defensible line than the Potomac.
While a popular cry might have been "On to Richmond!" I don't think McDowell was either stupid enough or imaginative enough to think that this was what his encounter with the Confederates would accomplish.
The goal was to keep DC safe and to keep Confederates from linking up with sympathizers in MD to create a pincer.
Also, Beauregard publicly said after First Bull Run that his goal was to take DC. Beauregard said a lot of things, of course.