Rapping has its origins in the African-American tradition of "toasting," which involves telling (sometimes improvising) a story in verse. One of the most famous is that of the Signifying Monkey. See Henry Louis Gates for his theory on the importance of signifyin (wordplay) in African-American culture.

If you want to look at the development of rapping (as opposed to hip-hop and DJing), then some important figures would be Gil-Scott Heron and the Last Poets (who incorporated spoken-word jazz poetry with funk/soul music) and James Brown (who is called the Godfather of Soul; his rhythmic innovation is that he had his band emphasize the downbeats (1) and the offbeats (2 and 4)).

Other theories involve the New York immigrants from Jamaica, who brought their tradition of toasting over rhythmic beats. An important figure is DJ Kool Herc, who combined the drum beats of people like James Brown with toast raps.

Rhythmically, rap music had much more in common with its (mostly) contemporary black musical traditions: funk, disco, soul, etc. Jazz has started to die down by the 1980's, with the exception of Miles Davis, who incorporated popular styles into his playing. While jazz was still strong then (and is now), it did not have the same impact as the previous parts of the century. The musical history of funk, disco, soul, etc go back to the R&B (or rock n roll) of the mid-20th century (for example, artists like Little Richard and Louis Jordan).

It's not the place of history to determine which art form better represents "the struggle." Many of these musical styles (blues, jazz, and rap) come from people who did not have many financial opportunities. When the blues was started to be a common form of music, it was because guitars were cheap. Jazz started from New Orleans brass bands, and few of the black players were wealthy (the piano, while still relatively cheap compared to today, did attract some more well-off musicians, like Fletcher Henderson and Duke Ellington; but the piano was more expensive and had the stigma of being feminine). In the 1980's, poorer black people utilized their voices and their record players, as musical instruments and training weren't cheap in the way that they were in the early 20th century. This is the financial situation that led to black musicians taking up the instruments that were available to them. (However, it's important to note that no genre has a completely monolithic musician makeup). They used the circumstances available to "represent the struggle."

A common theory is that the African-American musical tradition descends from the tradition of the griots throughout West Africa. Where jazz emphasizes improvisation and the rhythmic complexity found in West African music, rap emphasizes the lyricism and rhythmic pulse found in West African music. But remember that all of these have remained in African-American music for hundreds of years, and there are musicians in all these genres that have emphasized all of these musical elements.

Some musical examples:

Toasts

Malian griot playing kora with a calabash player

Louis Jordan

Gil-Scott Heron

The Last Poets (Lightnin Rod)

James Brown

Political blues

Political jazz (or a song that has lyrics)

Political rap

One thing about rap is that it is able to "represent the struggle" more explicitly than jazz. This is not due to some rule about jazz, but about the way that jazz musicians chose to express themselves. After Bebop and Charlie Parker in the 40s-50s, jazz became much more abstract. Pieces without singing became more popular. Most of the songs that jazz singers sang were "standards," which are typically Broadway/Tin Pan Alley songs, and are normally not politically-charged. Though two exceptions would be Summertime by George Gershwin and Strange Fruit (linked above). Though both of those songs were written by white men.

Also, remember that the main thesis of The Boondocks is one that is critical of the media's portrayal of black people. The characters (except Huey) are stereotypes of the media's black characters (with the zenith being Riley). So Martin Luther King would not be observing reality, but observing modern media's conceptions of black people.

It's very late and I'm tired, so please say if anything's unclear.

Firstly jazz and rap are cousins. They take african rhythm and weld it to respectively western harmony and language which is then improvised upon to create a narrative.

I'd say that OP could learn a lot from the work of James Brown. His music was in my view the most 'african' of any major 20thC artist. Percussion is everything.

And this is where I think the birth of jazz lies. Syncopation is the innovation that jazz brought to western harmony.

In turn that self same sophisticated harmony informed the musical structures of artists such as Marvin Gaye and Curtis Mayfield.

Culturally speaking it's important not to think of jazz as a stale dead form. The likes of Miles Davis and Charles Mingus, though dressed in suits were shall we say 'bad motherfuckers' (both their autobiographies are fantastic.)

I feel there's a dichotomy present in 'black music' of freedom and conformity. Freedom as in extension and passion (say john coltrane or gospel) and ruthless conformity to the groove.

(Hoping to come back and flesh out some of these points but getting ready for work now)

Unfortunately I don't believe this is an answerable question. Jazz/Motown is a rather arbitrary place to start. Start by looking at where Jazz/Motown came from. Black culture (edit: is there any sort of agreed upon definition of black culture!!??) is/was associated with Jazz/Motown but also funk/disco/R&B/gospel/blues(as you mention)/hymnal and all other sorts of genre(not to mention the problem of defining genre, similar to the problem of defining species in biology). My point is, it's gonna be difficult to find a clean linear progression that shows jazz to rap.

I also question whether rap dominates black culture, at this point it may be ubiquitous with pop culture. For example Run DMC and Drake probably don't have much in common these days. Neither does NWA/a$ap rocky/Rick Ross/atmosphere. At this point rap isn't even a specific genre. Also just as rap has evolved, Jazz/Motown has continued to evolve and isn't what it used to be in the 70's.

You bring up an interesting point regarding "wouldn't jazz/blues/motown be able to represent black struggle?" All I can say is that changes/evolves over time/geography. What speaks to people changes over time.

You are correct in that your interpretation of older music being "better" is skewed. It doesn't work that way. Is Olivier Messiaen better than Robert Johnson? I guess it just depends on your personal taste.