TBH, Nixon's Southern strategy appears to have been largely rhetorical. I mean, desegregation was primarily in the hands of the courts and Nixon's SCOTUS Justices, with the exception of Rehnquist, weren't exactly social conservatives either. Also, welfare cutting only gained steam later--not under Nixon.
Don't think SCOTUS is relevant here tbh. Southern strategy was based on capitalizing on white feelings about desegregation not about actually stopping it. That being said, Courts can't just desegregate at the snap of a finger, you need to keep suing over and over again after cases and controversy found to have standing that are successfully appealed from segregationist dominated state and local courts and the court then needs to file specific orders to stop it. Long and difficult process and the Southern state and local officials knew this and successfully held off desegregation for decades and many argue that a state of semi segregation still exists in practice. SCOTUS is a court consisting of nine people and a bunch of law clerks they can't micromanage this, only can interpret the law in the cases that come to their court, they are deferring to the executive branch to enforce the law as they interpret it.