Each of these questions is the subject of bookshelves of dissertations, scholarly articles, and books.
In a sense, they all require us to engage in alternative history–that is, historical fiction–to come to answers.
As I understand it, Brown v. Board of Education–result of the NAACP LDF’s legal strategy and extremely partisan behavior by numerous federal judges–spurred private activism, which itself ultimately brought segregation to Americans’ TV screens and led to adoption of the decisive VRA of 1965. That’s what ended segregation, not the 1964 CRA.
Each of the three had a role. There’s no way to know what would have happened without any of them. In general, however, it was Congress that ended segregation. Historians today tend to resist this conclusion, because they want to stress black agency–just as they want women to be responsible for women’s suffrage, even though of course it was all-male legislatures that adopted the amendment at the behest of all-male electorates in nearly all the states.