Horses for courses...
If you add a TC the lens *becomes* a different lens. i.e. w/ a 2x on my 400 f/2.8 it is literally an 800mm f/5.6... it has the magnification, FOV, and DOF of an 800mm f/5.6. The only difference is that it retains the minimum focus distance of the 400mm.
Everything is a tradeoff. If you choose to crop (or use DX/DX mode) you get to use a lower ISO, Faster SS, or a sharper aperture which may mean the final result will be "better," assuming you have enough pixels remaining. On the other hand, if you choose to use a TC/longer TC you will have to use a higher ISO, slower SS, or wider aperture. But you have more pixels to work with (down sample) and the final image could be "better," particularly for larger display.
I use TC's when the light allows me to also use an acceptable ISO and stopped down; usually the 2x III. After that it's time to start trading off... Usually I use the D810 and will step down in TC power to get better results (subject dependent)... when I get to DX crop required I switch to the D4 and higher TC power because the D4 does so much better in high ISO noise (at 1:1). Then I start stepping down the TC's again...
But it also depends on if I'm handholding (typically), can use flash effectively (seldom), where my SS is (1/ 2xFL min desired), what size I want, how badly I want it, etc, etc,...