The 500 mm PF with a 1.4x TC is f8. On a D500 or D850, that means that only a small number of focus points near the center of the frame work. Not sure about the D5. From what I have read, there may be a need to focus tune a DSLR + lens +TC combo, even if the bare lens does not require it.
On a Z7, all the focus point work across the frame, even with an f8 combo. Also there is often no need (or at least less need) to focus tune.
I find the image quality with my Z7 + 500 mm PF + 1.4x TCIII to be quite good. (Purely from an image quality perspective, I think it is also quite good on a D500 or D850, although you may need to focus tune.) I started using this combo to photograph warblers, as they are small and I could use all the focal length I can get. They move around a lot and quickly, but most of my shots are with them perched. I also used this combo a lot this summer to photograph loons, mergansers, herons, and other water birds. But I am not a professional and I usually shoot handheld from a canoe/kayak or while hiking. I don’t think a D5 + 500 mm f4 + TC would work for me in a kayak, although it would offer many benefits.
The TC does slow autofocus some on the Z7 + 500 mm PF, depending on the light, but I have done birds in flight with that combo too. Not the best set up, but good enough to be worth using when the opportunity arises, especially for slower birds like eagles, herons, egrets, swans and the like. Given the focus point limitation, I mostly don’t use the 500 mm PF + 1.4x TCIII on my D500 or D850.