What makes you think Wilson couldnt have won based in Virginia? He was perceived as a Southerner during the campaign. Being from NJ didnt change that.
It's not where he was from: it's where he had become prominent in politics. Wilson would not have been a likely successful politician in Virginia. Being a governor of NJ did not have as a pre-requisite staking out particularly fire-breating positions on race, religion, segregation, etc. Matters of importance for the South were never too politically important for himself.
Same, BTW, holds for Truman. That family WAS culturally fairly Southern. But a Senator from Missouri didn't need to pledge to fight for segregation till his last breath.