Having done research and written a book on Jimmy Claxton, the term "the first" is oversold. Very rarely is the popularly acknowledged "first", actually the case. Claxton was not "the first" either, just one of many unacknowledged pioneers in the sport.
Not to diminish Kenny Washington's achievements, without question a phenomenal athlete, but there were many other black professional football players in the 1920-30's including Fritz Pollard (1920), Bobby Marshal (1920), Duke Slater (1927, 1929), Joe Lillard (1932, 1933), and Ray Kemp (1933) just to name a few.
Here's Claxton with the semi-pro Edmonds nine, 1925 Snohomish County Independent Champions (photo courtesy of Mukilteo Historical Society):