All good discussion. I own a Sigma 500mm f4 Sport and a Nikon 500mm f5.5 PF. I have owned a Nikon 500mm f4.
Briefly, my conclusions are as follows.
All of them are sharp and focus well but the 500PF is obviously much more portable and hand-holdable. The other two can be hand held but not for long! I did not like the Nikon 500mm f4 that I had when mated with a TC1.4 and consequently used it little in this configuration. After I sold it, I waited a couple of years and bought a Sigma, largely based upon Brad Hill's analysis. I recommend reading this.
I have found the Sigma to be significantly better behaved with its own TC 1.4 in all respects than the Nikkor f4 was and have to say that the Nikon 500mm f5.6 PF does not match the Sigma when shot at 700mm with the TC 1.4. Of course, the Nikon is then a f8 lens. The Sigma by contrast, makes an excellent f5.6 700mm lens that is very sharp and focusses well. It is also extremely sharp and fast focussing at 500mm and is tuneable with the Sigma dock. You can buy these second-hand for very little money. Similarly, the Sigma TC1.4 is not expensive. Interestingly, you see very few second hand Sigma 500mm f4 lenses for sale when compared with the Nikon 500mm f4 but I suppose that more Nikons have been sold.
When I bought the 500mm PF, I considered selling the Sigma and am glad that I did not. I now use the Nikon PF as a travelling/hiking lens, sometimes with a monopod but very rarely with a tripod. I use the Sigma as a 'destination' lens, usually on a tripod but rarely walk more than a couple of miles with it or hand-hold it. If possible, I prefer the Sigma because of the f4 maximum aperture and much prefer it when using the TC 1.4. If I could only afford one of these lenses, the PF is obviously the more multi-role capable but they are both excellent. I do think that some people underestimate the significance of the f4 versus f5.6 maximum apertures and personally think that there are several important reasons to prefer the f4.
I have never used any of these lenses on a Nikon Z body and would think it possible that the Nikon PF might focus better at f8 on one of these than it does on a D850/D500.
All my own opinions of course and other opinions are available!