Sorry, I am a little late in replying. I have used the F-mount 1.4x TCIII, 1.7x TCII and 2x TCIII with my 500 mm PF on Z7 and Z7II bodies (and to a lesser extent, a Z6II). All the AF points work across the frame, even with the 2x TCIII, although that TC produces an f11 combination with the 500 mm PF.
(As others have noted, the new Z mount TCs do not work with F mount lenses. At this point, the only Z mount lens they work with is the 70-200 f2.8 Z lens. I find they work nicely with that, especially the 1.4x TC Z. They will presumably work with subsequent longer Z mount telephotos, which I hope we will start to see soon.)
In optical quality, the 1.4x TCIII is the best of the 3. But I find both the 1.7x TCII and 2x TCIII are quite good -- definitely good enough for me to find them very useful in my wildlife photography. And I believe generally better than cropping to get the same field of view.
I have used the Z7II (and previously the Z7) with the 500 mm PF and 1.4x TCIII or 1.7x TCII to photograph larger birds in flight -- bald eagles, swans, great blue herons, sandhill cranes, great egrets and the like. I do find that AF is slower with the 2x TCIII than with the other 2 TCs, so I do not use if for BIF.
I got the 2x TCIII earlier this year to use in photographing a great blue heron rookery this spring with the Z7II and 500 mm PF. The nests were anywhere from 85 to 140 yards away, so I wanted all the focal length I could get. I mostly shot the combination from a tripod, but occasionally hand held. I had thought about renting the Nikon 800 mm f5.6 with its dedicated 1.25x TC, as that would have been excellent optically. But the 800 mm lens is also quite heavy and would have been expensive to rent for the 3 months that I photographed at the rookery.
The other reason decided to try the 2x TCIII is that Brad Hill mentioned on his site that he liked the 2x TCIII with his 500 mm PF on a ZII body. You might look at his site for some examples of photos he took with that combination.
I agree that it would be nice to see some new Z telephotos with the Z9 announcement.