It depends on what you shoot and the ambient conditions that you shoot in. While VR is great to have, it won't allow you to use a faster shutter speed to freeze the motion of a moving subject. The 300mm f/4 also has a slight advantage in subject isolation compared to the 80-400mm.
I think the 300 f/4 AFS is a little sharper - its as sharp or sharper than my 200-400 which is sharper than the 80-400. The 300 f/4 is probably a little sharper with a 1.4 teleconverter, but it gets close.
What the 80-400 provides is flexibility - and it provides VR. So as an all purpose long lens, it is probably a better choice. Image quality is still very good - just not equal to a prime like the 300 f/4. The redesign of the 80-400 has created a remarkably fast zoom. While the earlier version - and most alternatives - move to the narrowest aperture pretty quickly, the new 80-400 only is an f/5.6 lens at 380-400mm.
I'd look at the rest of your kit and what you are trying to accomplish. In my case, I have the 70-200, 300 f/4, and 200-400. As a travel lens the 80-400 would replace all three of these lenses. It's perfect for someplace like Denali where space is at a premium - or on the water in a canoe. It's a great lens for hiking and portability. On the other hand, I'd probably prefer the quality of the 70-200 and 200-400 for places like the Smokies and Yellowstone. And I can use a range of options for places like the St. Augustine Alligator Farm depending on my choice of subject.
I agree 100%. I owned the older 80-400mm AF-D VR and I used it in variety of situations where I was photographing wildlife and needed a more portable option (including Denali and shooting from a canoe in the Everglades ). Unfortunately, I eventually got frustrated at its autofocus speed and sold it, with the 200-400mm, 70-200mm and 300mm mostly taking its place. The new one is quite nice, and I can see using it in those same kinds of situations again, including BIF shots, where it should be very handy.
I've tested the 300 f/4 against the 200-400 at 300mm and at 400mm against the 300 with 1.4 teleconverter. I found in both cases the 300mm lens was sharper. And the 200-400 is sharper than the 80-400.
It makes sense that a prime lens is sharper than a zoom. Most primes are sharper than zooms, and the telephoto lenses are all very sharp. It's really an unfair comparison.
But the 80-400 and 200-400 provide flexibility, and that is a significant advantage. Filling the frame or leaving enough space can turn an okay image into a much better image. I'd consider the lenses alternatives for different situations.
My guess is if you have both, the 300 f/4 will get a lot less use and the 80-400 would be your "go to" lens regardless of a slight difference in sharpness.
I think the 70-200 Ver 1 is the weak link in your kit. The vignetting and sharpness are weaknesses compared to the newer 70-200 on FX bodies.
When you do your test also make sure to compare the bokeh between the two. I agree that the 300mm F4 is sharper than the 80-400mm AFS even with the 1.4x TC, but I think that the bokeh of the 300mm is notably better as well.