I should point out that this list is an attempt to compile the most highly regarded courses for historical informational purposes and to spur further conversation and understanding only. This is simply my take on it based simply on a lot of personal historical research of the time period, and should be taken with that grain of salt.
I've only played a few of the courses in question, and some of them have changed significantly for better and worse over the years, but this is an attempt to classify them as they stood at the time in question, not today. With that said;
The Top Ten
Brown Deer Park - Milwaukee, WI - 1929
Cobb's Creek - Philadelphia, PA - 1916
Community GC - Dayton, OH - 1909
Eastmoreland - Portland, OR - 1917
Forest Park - St. Louis, MO - 1912
Griffith Park (Wilson) - Los Angeles, CA - 1924
Harding Park - San Francisco, CA - 1925
Keller GC - Minneapolis, MN - 1929
Rackham GC - Detroit, MI - 1924
Wellshire GC - Denver, CO - 1926
Other contenders? Here's some that come to mind.
East Potomac - Washington DC
Shawnee - Louisville, KY
Coffin GC - Indianapolis, IN
Rancho Park - Los Angeles, CA
Waveland - Des Moines, IA
Franklin Park - Boston, MA
By definition I'm not including resort courses, which were played primarily by the same upper class folks who populated the country club game at the time. Submissions of others for consideration is most welcomed. Thanks.