I can not comment on the CCT but I have ridden on the Mount Vernon Trail and I can say that was so bad I got off and rode on the Parkway to Mount Vernon.
I think trails like these are more for sightseers than cyclists, and sightseers being people on bikes taking in the sights. Way too many walkers, joggers, people walking their dogs, roller bladers, kids playing and such.
Yeah you can ride a bike there but expect to go 8 mph.