I find that I really can't break 17 mph unless I ride more aero, but then it's also somewhat windy in this season. I really pump it and usually run an average of 19-20 mph at full speed, riding in the drops, and accelerating on the hoods. Flat bars just don't cut it for speed.
Have you considered buying a used vintage roadie with drops? It could be a very inexpensive buy if you keep an eye out for yard sales and the like.