You are absolutely correct. For reasons unknown to me the south often gets blamed by yankees for some of our worst liberal presidents. On this very forum I've seen us blamed for FDR, for Woodrow Wilson, for LBJ, for Clinton, for Truman, you name it. It's always "if the south hadn't voted Democrat blah blah blah" but the fact is when you consider each and every one of those elections it is dubious if not outright fraudulent to blame the south as they do.
Woodrow Wilson? Yes, the south voted for him but the reality is he got into office due to TR and Taft splitting the Republican vote.
FDR? The south voted for him...but so did practically every other state in the country! And as you pointed out, he actually ran on a platform of cutting government spending in 1932!
Truman? Last I checked the "solid south" was not quite so solid from about 1948 on, with several states going for conservative Strom Thurmond and the next time around several southern states actually voted for Eisenhower.
1960? Nixon won 4 CSA states/territories. Mississippi and Alabama cast protest votes by throwing their support behind conservative democrat Harry Flood Byrd.
LBJ? Hardly. The ONLY states that Goldwater won were all former members of the CSA. So much for the solid south that time as well.
1968? Nixon and George Wallace split the south with Humphrey winning only one southern state.
And of course Clinton. Never did he win a majority of southern electoral votes, despite being a southern governor himself. Yet Clinton swept every single yankee state but Indiana not once but twice.