I've had the 80-400 since about 2004. The whole front end of the lens barrel moves back and forth when it focuses. This movement is driven by the AF motor in the camera body and because of the weight, the mechnism is heavily geared, so it is very slow. the lens also hunts for focus unless the subject is clearly visible - i.e. it overshoots, racks all the way out and then back in, looking for the subject.
For an airshow you need fast AF to acquire and track the aircraft, especially fast movers. So as much as I love the 80-400, I think you are much better off with an AF-S lens to ensure fast AF speeds.
The 70-200 f2.8 will be great, but on a DX body you may find that a 1.4 TC is long enough. Remember that you lose 1 stop with 1.4TC, 1.5 stops with 1.7TC and 2 stops with 2.0TC. As you lose stops, the AF becomes progressively slower too.
If you were considering investing in a 80-400 for wildlife photography, I'd say go for it, but not for airshows. For similar kind of money you will find Nikon's excellent 300mm f/4 AF-S, a Sigma 50-500 or a used Sigma 120-300 f2.8 OS. All of these should offer better AF performance than the 80-400 and similar or better image quality. both the 300 f4 and the 120-300 f2.8 will work nicely with a suitable 1.4TC so that gets you to 420mm, similar long-end focal length to the 80-400