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School desegregation, while important to the NAACP, was far from the most important or most emphasized program they followed for cultural equalization. Remember that the NAACP was founded in 1909 and was a nationwide organization with hundreds of chapters with different, yet similar, goals. Throughout the years the NAACP pushed for many different cases, anti-lynching legislation in the 1910's-1930's, Military desegregation in the 1940's, bus desegregation with the prime example of Rosa Parks and the Montgomery NAACP leading the charge in Alabama. They also tackled restaurant segregation, bus stations, public parks, police brutality, housing laws, equal pay, voter registration and many more. While the importance of Brown v. Board absolutely cannot be understated, School Desegregation wasn't their only goal by far.
Others have explained how the NAACP had a far broader approach than just education, but no one has explained why education itself was seen as so important.
Education was a benefit which had been denied to African Americans across the South since the Civil War. White populations preferred to have uneducated African Americans who would do their work. Voting, a supposedly equal right, required tests to be passed to register. Uneducated people had a hard time passing a test where they had to read something like the Constitution with no training in literacy. Things that we see as basic life skills, like reading a cheque, or filling out forms, and managing finances, were harder for those who were never educated and made them prime targets to be taken advantage of. The fact that they were black merely made it a certainty that there would be no repercussions.
Education helped African Americans with financial autonomy, political expression, and even intangibles, like confidence and self-belief. Think about how proud a child is to write their name, or an adult is when they have perfected a new skill.
Education also fed into an older narrative of community autonomy. Schools were spaces where the next generation was learning about the world. But in the early 20th century, the African American students were often taught by white teachers, who weren't teaching black history. It was a form of erasure. NAACP efforts in South Carolina for example, focused on getting black teachers in these schools, and equalising the funding. This was key in teaching African American history to the current generations.
Adult education was hugely important too, because it worked to fix the discrepancies left behind by inequality in the public schooling systems. Adult education was a communal practice stretching back to the days of slavery. In the 20th century, it came to be a meeting of members of the community who would work together to learn how to read and write. The communal aspect meant that children learned alongside parents and grandparents, and everyone was exposed to the experiences of others in their community.
The biggest program of this sort was the Citizenship Education Program which aimed to educate African American adults to a point where they could register to vote. It helped to form the grassroots base of the classic Civil Rights Movement.