You can debate this stuff forever. As I said before a good total body workout once or twice a week is going to help you stay fit to play tennis. So it's nice to do. Because its an overhead sport I would add in rotator cuff work.
But if you REALLY want to improve your serve. The key is to umm PRACTICE THE SERVE. Better yet practice the serve with coaching and video analysis.
It's the same for baseball players, football players etc. Tom Brady is not a weight room junkie haha. Tim Tebow is. Who throws the better ball?