I think the 1.4x TCIII works quite well with the 500 mm PF on a Z body. I have used that combination extensively on a Z7 with an FTZ. Of course, there is some loss of image quality, but I think it is less than doing a DX crop, and the resulting image is still quite good optically. (I think that was also Steve’s general conclusion in a video or post he did a while back on TCs.) I have tended to use that combination on my Z7 and not on my Z6 (now replaced by a Z6II).
You do lose a stop of light, as the combination is f8. Not ideal, but I have not found that to be a big problem practically most of the time, as the Z7 is pretty good even at ISO 6400. (I spent much of the summer shooting waterbirds and wading birds with this combination from a kayak, with light levels varying a lot. I also shoot warblers, where the extra reach is very useful.) And Topaz DeNoise AI or DxO Photo Labs Prime and Deep Prime noise reduction is very helpful when needed. Of course, f8 also means less ability to isolate a subject.
All the focus points continue to work at f8 on a Z7 or Z6 (and I assume also the ZII models, although I have not tested that yet). That is a significant advantage over the D500 or D850, where only a small number of central focus points work with an f8 lens. In addition, there is likely to be little need to focus tune the combination on a Z body, something that might more often be needed for optimal results on a DSLR.
Focus speed probably slows down a bit, but again I have not found that to be a big problem in most cases. I have even shot BIF with the Z7 + 500 mm PF + 1.4x TCIII — both larger, slower birds (eagles, swans, egrets and the like) and smaller faster birds (osprey, gulls and swallows).
I have enjoyed using the 500 mm PF on a D500, D850 and Z7, and tested each of those bodies with the 500 mm PF and the 1.4x TCIII. Based on that experience, when I want to use the 1.4x TCIII with the 500 mm PF, I choose the Z7. Other than pixel density, I would expect the Z6II would work as well, with probably a bit better autofocus and better low light performance.