This site shows the geographic distribution of theaters in the United States in 1902 and 1913.
Data derives from booking agent Julius Cahn’s Official Theatrical Guide, Vol. 7 (1902) and the The Julius Cahn-Gus Hill Theatrical Guide, Vol. 17 (1913). Cahn’s directory undoubtedly undercounts US theaters for various reasons, but the general shape and distribution of theaters is correct.
I have produced a similar map focused on the Theatrical Syndicate’s operations.
Populations from the Cahn guides; railroad mileage from Statistical Abstract of the United States, 1912.