BTW, I also suggest the 70-200 2.8. With a teleconverter, I'd think it would be long enough (and fast enough) for wildlife.
You should really only add a TC when you *have* to, not every single time, since you lose image quality when you do so. Moving wildlife, especially raptors are already a challenge, and even large animals like Elk don't often let you approach. Longer is almost always better for wildlife, and always better for birds.
I shoot *lots* of wildlife, and *lots* of birds a good chunk of them *large* birds- I can tell you the number of times I've thought "Gee, I wish I had a nice 200mm lens!" or "Hey, 280mm would be great here!"- probably six times in the last ten years (most of those when a Bald Eagle was flying overhead and I couldn't get the monster off the tripod fast enough for a hand-held shot-- and really I was more thinking "Gee, the 200-400VR is a nice cheap, light lens, I bet I could get a great shot right now if I had one...")
In fact, I'd say the only general situation[1] where I thought the 400mm was too much lens with a non-dog animal in the viewfinder was at the Elephant compound at Washington D.C.'s National Zoo (and only the Elephant compound- for the rest of the zoo it worked really well- I just wish I could sell the images without encumbrance.)
I've owned the Nikkor 80-400VR and the Sigma 50-500VR as well as the 300/4 EDIF and 400/2.8. Going back through my thousands of wildlife images, the number of times I wasn't at 400mm or 500mm with the zooms is less than 1% of the time, and I can probably count the times I had to back away with either prime on one hand wearing mittens with my thumb tucked in.
Now, you may be one of those folks who thinks a 2x TC is an acceptable compromise, but I'm guessing that the "I'd think" in your above statement means you don't often shoot wildlife and probably don't own the 70-200 with a TC- in other words, it looks a lot like you're making a ~$2000 guess. I think that does the OP an injustice.
I think a 1.4x TC gives you good IQ when you absolutely need longer (that is, I'd prefer to shoot without it, but wildlife may not give you that luxury- and remember I'm shooting off a 400mm *prime* base platform, not a 200mm *zoom* one) and I think the 1.7x TC is marginal, but again sometimes you need that reach. Personally, I never shoot with a 2x TC, I find the loss in IQ unacceptable for my work in general (I've seen some great shots even with stacked converters, but find that even with a pristine prime, I'm not happy past 1.7x, and I'm grumpy when I have to go there)
With a 1.4x TC, the 70-200 gets to be 280mm at f/4- while the 300/4 gets to be a 420mm f/5.6 so you start out better (more length, same speed) and end up much better (the sort of length that's good for wildlife.) If you could routinely get good shots at 200/2.8, the 70-200 would be a good deal because you'd have longer to shoot at the beginning and end of each day when all the animals are out eating, but in my experience that's not a realistic expectation.
The 70-200 is a great lens, but it's not a great wildlife lens.
[1] Back when I shot film there was one juvenile Great Blue Heron who'd cause me to back up occasionally when shooting with a 210mm lens on a 645 body (around 130mm on a 35mm.)