Using extensive Atlanta based stats and looking at all wintry precip for all winters since 1878-9, I 100% agree with you on the lack of correlation between large scale liquid equivalent precip and wintry as I found essentially none. However, I most certainly found a pretty significant correlation between DJF temps and wintry precip: 67%/50%/33% of cold/near normal/mild winters had above average wintry precip in Atlanta.
As I have said here before, precip isn’t important but temperatures are as there is twice as much chance for above average wintry precip at Atlanta in cold vs mild winters based on looking at about 140 winters. Then again, that’s still nowhere near a 100% correlation. It qualifies as a significant partial correlation though.