You have a D600 and D7000, and I guess you are thinking about the D7100, or perhaps a D400 if it ever happens- something with maybe better AF and maybe better res than the D7000? You have a 70-200/2.8 and a TC17 so you are basically limited to 340mm. Do I have that right?
Since you mention the 300/4 AFS, and I own that, as well as the TC14 and TC17, I have some thoughts in line with your discussion of your shooting style.
The 70-200 + TC17 is easy to shoot hand held, as these things go. Plus it has VR to fall back on when you are under 1/500s but perhaps that is not important based on your last post.
Personally I shoot the 300/4 + TC17 hand held only on my D700 (FX). On my D300 I shoot it with the D300. That difference in field of view is critical for me. While I can shoot the lens at 420mm on DX or 500mm on FX, with some skill and daring, I find it "impossible" to shoot at 500mm on DX. Not really impossible but not viable.
Now, I do get very good results at 500mm on DX, but only on a tripod or at least a monopod. Otherwise I cannot even keep my framing under control. That is me personally but I think I am fairly good at it.
I think, for the reasons above, if you go the 300/4 route you may be surprised at the increased difficulty of those increased focal lengths. And if you do not already have a TC14 your should consider it or at least understand that you may find the TC17 too much to handle, especially hand held.
Back when the 300/4 was my longest lens I always used it on a tripod. When I got a 300/2.8 it went on a shelf but I never sold it because I respected it too much, and around the 300mm focal length the 70-200 got most use as a hand held setup.
I went back to using the 300/4 when I learned that even with a great gimbal (Wimberly WH-200) there were many in flight situations that are simply impossible to get with a gimbal. For example, it is impossible to get a straight overhead shot, nor is it possible to swivel around more than 90 degrees or so while tracking a bird. So you are correct that for in flight shots a hand held camera can be a better solution.
Where in flight shots work on a gimbal is for birds flying reasonably low to the ground and steady, and rather slowly. I have thousands of Snow Goose in flight images captured on a gimbal but that is an ideal subject for that, and the environment (Chincoteague NWR) is also ideal (on a rare good day in the migration season).
For 500mm DX, at a minimum you will likely need a good monopod and head. That may not help much for in flight shots but for stationary shots it will do wonders for your consistency and framing.
I find the 300/4 + TC17 to deliver amazingly crisp professional level images even wide open. But it takes considerable skill and daring, a lot of luck and multiple frames (or very high shutter speeds). And that is why I stress the support.
I don't shoot anything denser than 12mpx DX but I suspect that if you are looking at the D7100 or similar for the 24 mpx then you may be disappointed with the results with the 300/4 at 500mm unless you give it some serious support and apply some very good technique. I think it takes a virtually perfect image to take advantage of those extra pixels.
This is an earlier hand held shot of that Hermit Thrush, with the D700 and 300/4 + TC17, at f/9 ISO 1600 1/1000s. However, I was sitting on the ground to get a better angle and a little more stability. I knew I had an unusual opportunity and wanted to get the best shots possible.
The setup was about as good as it ever gets for small birds, in good light and an unusually cooperative bird. If I could do that more often I would use that lens more and my 500/4 (plus flash) less often. But this is not the norm. It does show what that lens and TC can do, though.
That is about comparable in every way to what I get with my 500/4 and TC14 on a D300. I just had to get much closer than I usually manage.