I have the relatively new 70-300 AF-P FX lens and like it a lot. I have used it on a D500, D810 and D850 when I want a light-weight telephoto and don't need more than 300 mm. It focuses quickly and accurately, including for fast moving birds and other animals. I have also used it for handheld scenic landscape shots while hiking. The image quality seems very good. Thom Hogan's review says the AF-P version is better optically than the older AF-S version. I had the older AF-S version in the past and this conclusion seems right to me, although I did not own them at the same time and thus did no direct comparisons. Thom includes the AF-P FX version in his list of lenses recommended for FX bodies, including hi-res bodies like the D8xx models.
My wife gave me a Z7 for Christmas and I have tried the the 70-300 AF-P FX lens on the Z7 with the FTZ adapter. The lens seems well matched to the Z7 in size and weight, even with the adapter. I think the Z6 is the same, or roughly the same, size and weight as the Z7, so this should be true on the Z6 as well.
So far, I have used it on the Z7 just casually, walking around neighborhood, taking pictures of trees, houses, squirrels and a few perched birds and only using single point autofocus. In this mode, the camera and lens combination seems to focus fast and accurately. The image quality seems the same as on my D850. I have not yet tried it on fast moving birds and other animals -- I intend to keep my D500 and D850 for wildlife photography in general -- although I will probably give it a try some time for that.
So I think the 70-300 AF-P FX version would be a good choice for the Z6, particularly as the Z mount lens roadmap does not show a native 70-300 at this point.