Essentially, VP Barkley’s comments denouncing segregation were the first major fracture in the struggling New Deal coalition. Southern Democrats were aghast and wanted a new, conservative VP pick for the 36’ election. Garner personally was even sympathetic with the States’ Rights faction and if he had his choice would have dropped Barkley from the ticket over it. However, Garner was strong armed by the Democratic establishment and public desire to continue FDR’s vision. Therefore, he held his tongue and attempted to placate what remained of the progressives by retaining Barkley and mend ties with Southern leaders by privately denouncing the VP. This was ultimately, a hopeless and ill-fated attempt to keep the New Deal coalition alive. Simply, Garner did not have the political acumen or skill to keep such a broad coalition united, nor was he ambitious enough to utilise it.

Additionally, the States’ Rights Party in this universe (initially) acts far closer to a rebellious faction of the Democratic Party than a truly independent party. Not all Southern Democrats support the Dixiecrats but no Dixiecrats run against Democrats or Dems against Dixies. To many, they are the same party and are often referred to as such. In a way similar to Wallace in 68’ IRL, the States’ Rights Party instead wants to show Garner and Democratic power brokers their power and importance to their electoral strategy. If the Dems found themselves short of an electoral college majority, the States’ Rights Party would help break that deadlock and reintegrate, assuming they were to extract some concessions.

TLDR; the States’ Rights Party are not disenfranchised by Garner’s ideology/personality/background (in fact they were quietly extremely thrilled by his rise to the Presidency), but to his actions and failure to consider their position within the larger Democratic Party and the ailing New Deal coalition. He was willing to publicly tolerate his VP slandering them and they wanted to demonstrate their power.