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For sports, you cant beat the superior focusing of the D3s. For indoor sports the superior AF system in the D3s and the better low light response is essential. For any sports, the frames per second of the D3s will give you much better shots.
d600,d800 just are not the right tool for the job
This sounds like someone that has never used a D600 or D800 for sports. Sure, sports is not the focal point for either camera, at least not at a professional level, but the D600 (never used a D800) is plenty adequate for sports. If money were not an issue, then by all means get a D4 and be done with it. I assume it is or the OP wouldn't be asking the question. Having used a D3 in the past, I just don't see it being worth the added cost. For me, the marginal improvement in AF performance is not worth the added cost. There are lots of nice lenses that I would rather put that money toward.
As for the FPS, I think 5.5 FPS is plenty to capture the speed of gymnastics or dance. More is always better if for no other reason than to have the choice, but is it $3000-4000 better? The number of FPS really has more to do with predictability and the speed of the sport than a requirement for good shots. The number of times I ran a D300 w/grip or D3 at full speed was very few. Sure, 8 FPS is great for professional football in decent lighting where the action is fast and unpredictable, but in poorly lit gyms you will struggle more to get your shutter speed high enough to stop the motion than fire enough frames to catch that perfect dismount. In addition, dance and gymnastics are predictable, so you can anticipate the shot better to capture that perfect moment.
patiently wait for the d400
I have to wonder how much shooting you have done in poorly lit gyms, because there is no way I would opt for a D400 over a D600 or D800 for indoor sports. Unless Nikon pulls a rabbit out of the hat, it is highly unlikely that the D400 will have good enough high ISO performance for indoor sports. It may be adequate to ISO 3200, but I would be shocked if it was as good as any of the FX sensor cameras. You are trading AF performance for high ISO capability. What good is marginally faster AF if the pictures are blurry because you can't shoot at high enough shutter speeds? In something like gymnastics, you could manually focus to a spot and still get images good enough to print (which you would never have to do with a D600), where as an image with significant motion trails will not.
One thing that I didn't see mentioned is the benefit of the additional resolution. It is nice to have those extra MPs to crop. Take something like uneven bars. It is easier to take a looser frame of the gymnast and let the camera focus track with in the frame, rather than take a tight frame and pan around as the gymnast moves around. You will appreciate those extra MPs to crop to something tighter when you capture that perfect moment. A lot obviously depends on your skill level. If you are comfortable panning to follow the action, then this significantly reduces the benefit unless you plan to print posters.