The court-packing plan in OTL did much to crystalize the "conservative coalition" of Republicans and (mostly southern) conservative Democrats that was to hamstring the New Deal for the remainder of FDR's years in office and
afterwards. I stated in a soc.history.what-if post some years ago that "that coalition would have eventually formed anyway. The near-unanimous support FDR got from Congressional Democrats under the emergency conditions of 1933 could not be sustained forever." I still think this is true, especially since the 1937-8 recession would weaken FDR's hand, yet without the bitterness of the court-packing fight, even conservative Democrats might be more willing to compromise with the administration. Furthermore, the court-packing fight was one major reason for FDR's subsequent blunder of the attempted "purge" of anti-New Deal Democrats in 1938, which further embittered conservative
Democrats and perhaps led to the New Dealers losing more heavily in the 1938 election than they would otherwise have done.