There were many influences tugging on JFK to engage the civil rights movement. Shriver was but one of them. Conservative Republican Senator Everett Dirksen was another.
More germaine to my point then who was first to pull JFK into the struggle, was my point that he was not a liberal, and at best, a reluctant participant in the civil rights movement.
Kennedy's Domestic Agenda
It is still possible to argue, as some leftists do, that while JFK might have been less than progressive prior to assuming the Presidency, once he was in office, he became the champion and hope for liberal-progressive. But this idea is likewise not borne out by what JFK said and did during his thousand days. Once again, it was the image of a "Vital Center" Democrat that prevailed, and more often than not leaning more center and right than left.
On Civil Rights, JFK conducted a policy that was virtually a carbon copy of the one Dwight Eisenhower carried out. Like Ike, JFK believed in the moral correctness of integration. Like Ike in the Little Rock High School crisis of 1956, JFK was prepared to use the power of the federal government to uphold the law, as he did when he sent troops to protect the admittance of James Meredith to the University of Mississippi, and later to more peacefully force integration at the University of Alabama.
But like Eisenhower, JFK also felt that the momentum for civil rights and integration had to be kept at a gradual pace, lest a situation of unrest and backlash erupt all over the south. Like Eisenhower, JFK had no great love for the overt activism of Martin Luther King and the SCLC or the Congress on Racial Equality, and frequently wished that the Civil Rights organizations would act with more restraint.
In the Spring of 1961, CORE began its infamous "Freedom Rides" on Greyhound buses from Washington to New Orleans in an effort to test whether bus facilities were being desegregated. Along the way, there was a great deal of violence, with many racists assaulting the riders and burning some of the buses. To protect the riders, JFK decided that some federal marshals would have to be sent along. But as Harris Wofford, JFK's civil rights advisor recalled, JFK was furious with CORE for inviting trouble, especially at a time when JFK was preoccupied with the upcoming Vienna summit with Khrushchev. "Can't you get your friends off those goddamned buses?" he angrily asked Wofford, "Stop them."
As the rides continued, both JFK and RFK were still upset by what they felt were the "giant-pain-in-the-asses" at CORE who had invited the trouble with the Rides. JFK felt that the more he had to openly side with civil rights, the more difficult it would be for him to get anything past the racist Southern Democrats in Congress who wielded considerable power. JFK wanted to be supportive of Civil Rights, but he wanted to see the movement act on his own terms.
JFK's less than wholehearted feelings of affection for the movement would surface again two years later when both he and RFK would agree with J. Edgar Hoover that King needed to be wiretapped because at least one of his advisors had suspected communist ties, and both JFK and RFK had met with King urging the civil rights leader to drop those men from his group. King refused.
Likewise, when it came time for King to hold his famous March on Washington in 1963, Kennedy's support was passive and tepid. JFK regarded any march as something that would only be counterproductive in efforts to get civil rights legislation through Congress, and tried to talk King out of it. Again, Kennedy was not willing to go out on a limb and give full 100% backing to the movement.
http://mcadams.posc.mu.edu/progjfk4.htm