I don't think that the issue is the difference between 1/200 sec and 1/250 sec. That difference should mean a tiny bit more subject motion - nothing more.
If you've missed focus, nothing will help. If your subject is moving rapidly, regardless of the 1/focal length rule you need to be able to freeze action adequately. VR has nothing to do with subject motion or missed focus. If you have it set correctly it can help with intentionally pan blurred shots.
The AF of the D600 is quite good - but with fast action you might get a lower keeper rate than the D800 or D4. Still - it should be able to do the job.