Here's a good example of "wait" that I am pretty sure I came across in Steven Levingston's Kennedy and King that addressed school integration. It is early in King's career but certainly not first.

President Eisenhower seemed to support integrating schools in steps. First, integrate graduate schools. They're all grown adults and had lots of international diversity anyway; a tiny addition of African Americans would make little difference. In 1950, grad school would have also been mostly men, so the fantasies about white women being put at risk is less of a factor. Once everyone calms down, integrate undergrad colleges, again with a pretty tiny number of Black kids whose segregated schools (or already-integrated Catholic parochial schools) could prepaire their students for college level work. A few, or several years later, THEN integrate the lower grades after we have a lengthy track record of claims of "the sky is falling" being proven ridiculous.

While I can imagine an adequately ethical White person of my grandparents' generation thinking this is a good idea, this was too much "wait" for King.

Bonus - in the 1960 election, Eisenhower's VP Richard Nixon ran against JFK. MLK Jr. supported JFK, but MLK Sr. and Jackie Robinson remained with the party of Lincoln and supported Nixon. Eisenhower/Nixon had given some limited, tepid support on a few issues, and both parties had substantial racially retrograde factions. The large Northeastern cities that allowed the Black vote were largely Republican, so things could have gone a few different ways.