JFK was involved in the race issue since before being president. In fact, it is possible that he won due to the Black vote, after he and Bobby had MLK, Jr. freed from jail.

https://www.youtube.com/watch?v=bH6RUxGgCPk

Fast forward to hour 1:36-1:39 and 1:43

If he had pushed too much, he would have lost the Democrat congressmen from the South. Keep in mind that those days, the south Democrats were the racist ones. Today the racists are in the Republican Party.