It's worth noting that the premise of the original question --that Shirley did not write down any music, and wasn't recognized as a composer --is incorrect:
Don Walbridge Shirley was born January 29, l927 in Kingston, Jamaica... At the age of nine he was invited to study theory with Mittolovski at the Leningrad Conservatory of Music, and he later studied with famous organist Conrad Bernier and studied advanced composition with both Bernier and Dr.Thaddeus Jones at Catholic University of America in Washington D. C.
...In l946 his first major composition was performed by the London Philharmonic orchestra.
...He has written symphonies performed by the Philadelphia Orchestra and the New York Philharmonic, and has composed four organ symphonies, numerous pieces for piano, two string quartets, and a piano concerto. He played as soloist with the orchestra at Milan's La Scala opera house in a program dedicated to Gershwin's music. Only two other pianists have performed there as soloists--Rubinstein and Richter.
With all that said, there are two separate racial/cultural dynamics at work here with specific regards to the pieces you've highlighted. First, there's a different relationship to preexisting work in all the musical traditions of the African diaspora than in the European tradition. Musicians working in an African diaspora tradition --which includes all the black American musical forms, most notably jazz, ragtime, blues, rock, gospel, soul and hip hop --tend to treat music as a ongoing conversation, meaning they bring their own creativity to bear on existing songs, rather than seeking fidelity to the source. Black music often quotes and adapts older work, but doesn't reproduce it, and even successive performances by the same performer may be different each time. On the other hand, in the European music tradition, fidelity to the source is prized, so in general performers try to reproduce at least the original notes and rhythms exactly as written. Because of this, it's easier to assign credit for a given song directly to a single composer in the European tradition.
It's also indisputably the case that black artistic contributions have historically been undervalued due to pervasive racism in the cultural mainstream establishment. The work of black artists has too often been viewed as a free natural resource, something that only gains mainstream value when adopted, adapted or flat out arrogated by a white artist. It's hard not to see racism in the reasons why Shirley saw limited success in classical music, and, like Nina Simone, had to redirect his work towards the jazz audience, performing more often in nightclubs than in concert halls. Both dynamics come together in jazz, a hybrid of African and European musical traditions and conventions. Due to the collaborative and improvisational nature of jazz it can often be difficult to assign a single definitive author to any given piece, especially given that many of the most iconic works are jazz variations on musical themes that originated outside the jazz world. By convention, however (and again, at least partly because of pervasive devaluation of characteristically African modes of creativity) the attribution is often given to the original songwriter, even, when, as in the cases of Shirley's work, it is clear that it has been entirely reimagined, in a way that fully qualifies as original composition by any standard.