Booker T. Washington was the first leader of the Tuskegee Institute, and by partnering with wealthy donors, helped to build public schools for black kids. He urged blacks to work hard and become educated.
I do not think Booker T would have liked Obama at all. There are two major strains of American Black philosophy
Those who live their lives akin to the self-reliant attitude espoused and promoted by Booker T Washington. Clarence Thomas is a man in that line of self-mastery.
Yet the other strain is derived from the perennial weed of grudge bearing as developed, marketed and propagandized by WEB DuBois.
DuBois established the line of Black communists that after three generations begat Barack Obama. Obama is the direct spiritual inheritor of DuBois.