To what extent was Jesse Jackson's emergence as a nationally prominent political figure a reflection of broader trends in African-American politics?

It's difficult to separate African-American politics from the rest, because black history is American history. It's true that there are African-American political leaders, and that the black experience of American history is radically different than the white experience of it, but there aren't obviously-different African-American politics. That's important for understanding the rise of Jesse Jackson. He didn't attempt to draw a distinction between black and white politics. He simply used politics to his advantage in ways that more radical African-American leaders didn't. In this way he pre-figured the rise of Barack Obama.
Jesse Jackson's activism didn't reflect broader trends in the African-American experience of politics, because he shaped them as part of its mainstream. He has never been a particularly radical figure, not when compared to leftists (like Huey Newton or Angela Davis) or conservatives (like Zora Neale Hurston or Clarence Thomas). He didn't define trends; he rode the wave.
Reverend Jackson has been very effective, however, at leveraging the power of praxis, as Antonio Gramsci put it. Whereas more radical black activists pushed their agendas to further an ideology, Jackson became famous for taking advantage of circumstances as he found them in order to help people and to raise awareness of injustice. Good examples of this are his work with Operation Breadbasket, the Southern Christian Leadership Conference, and his Rainbow Coalition. Those efforts were notable for their embrace of social change through community action and for their willingness to engage white political and cultural leaders.
If you want to understand what you call African-American politics, you should read widely about politics and history at least since the end of the Civil War. Pay particular attention to themes with strong influences of race and social justice, like housing and public services, voting and representation, and poverty and wealth inequality. There are myriad others, but those are good places to start. Read a range of African-American points of view, from Harriet Tubman, W.E.B. DuBois, and Marcus Garvey. Read Paul Robeson, Harry Belafonte, Eldridge Cleaver, and Angela Davis. Read Martin Luther King, Malcom X, and Huey Newton. Read Cornel West, Henry Louis Gates, and Ward Connerly. If you do that, you'll get a sense of where Jesse Jackson fits.
https://www.chicagotribune.com/topic/jesse-jackson-PEPLT007437-topic.html

https://www.nytimes.com/1987/11/29/magazine/jesse-jackson-aims-for-the-mainstream.html

https://www.pbs.org/wgbh/pages/frontline/jesse/speeches/jesse88speech.html