Because all that would do is deliver the White House to Nixon, and they have nowhere to go. Keep in mind that LBJ had his liberal allies too, such as Humphrey, Jim Rowe, Orville Freeman, etc. Civil rights was not really an issue in the campaign until MLK's arrest, and the dual phone calls, made it an issue. But as a policy issue in and of itself, JFK and Nixon said virtually the same things on CR. It was not a serious threat, more like a kid who doesn't get a lollipop and throws a temper tantrum. IOTL they were very suspicious of even JFK's liberalism, because it was not their standard-issue New/Fair Dealism. ADA et al. were in their own way as ideologically narrow-minded as Teabaggers. If you're not their type of liberal or conservative you're not a liberal or a conservative. Labor has more substantive grievances given LBJ's vote for Taft-Hartley, but they are not voting for Nixon, who was one of its principal authors in the House and campaigned in the '58 midterms on union-busting.
The South in 1948 had grievances that were very real from their POV: the beginning of the end of Jim Crow, which the more realistic of them knew would eventually happen. By then it was just a matter of delaying the inevitable. That was why they were simmering against FDR by the 1944 election (including the short-lived Texas Revolt, destroyed by Rayburn and LBJ): the GI Bill of Rights was an indirect yet massive assault on Dixie by allowing black servicemen the right to a post-secondary education.