In August 1999, Barack Obama strolled amid the floats and bands making their way down Martin Luther King Drive on Chicago’s South Side. Billed as the largest African-American parade in the country, the summer rite was a draw over the years to boxing heroes like Muhammad Ali and jazz greats like Duke Ellington. It was also a must-stop for the city’s top politicians. Back then, Mr. Obama, a state senator who was contemplating a run for Congress, was so little-known in the community’s black neighborhoods that it was hard to find more than a few dozen people to walk with...