Actually Washington (DC) was predicted to become a major industrial and commercial center. This is why many cities fought for the distinction. However it was too close to Baltimore, which was until around 1950 one of the 10 largest cities in America. In fact for many years Baltimore was second to NYC in population.
But Washington was always supposed to be a center of industry and commerical trade. Just didn’t work out.
Many capitals were chosen for their geographic centerness, such as Pierre South Dakota, Indianapolis and Columbus.
In Kentucky Frankfort was chosen for it’s midway location between Louisville and Lexington.
State capitals by in large are big within their state. Even the smallest State capital Montpelier with about 8,000 is large for the state. Remember Burlington the largest city only has 39,000.
Large state Capitals include
(Est 2002 Population US Census est) [overall rank city population] *=Largest City in state
Phoneix* (1,371,000) [6th]
Indianapolis* (783,000) [12]
Columbus* (725,000) [15]
Austin (671,000) [16]
Boston* (589,000) [20]
Denver* (560,000) [26]
Nashville (545,000) [27]
Okalahoma City* (519,000) [29]
Sacramento (435,000) [38]
Atlanta* (424,000) [41]
Honolulu* (378,000) [45]
Raleigh (306,000) [58]
Saint Paul (284,000) [61]
Lincoln (232,000) [75]
Baton Rouge (225,000) [79]
Madison (215,000) [83]
Montgomery (201,000) [92]
Des Moines* (198,000) [95]
Richmond (197,000) [97]
So state caps aren’t by in large small. 25% of the Largest 20 cities are state capitals. 22% Of the top 50 cities are state capitals and 19% of the top 100 cities are state captitals