We have a few variables here that bear consideration:
Lakeman posted he has a Z6. I used my 200-500mm lens on a number of Nikon DSLR's from 2016 to 2020. First a D7000, followed by D7100, then D7500. None of those cameras are in the same league as a D500 or D850 in terms of tracking fast-moving objects. In 2020 I decided to go mirrorless with the Z6 - I was just smitten with the advantages of the mirrorless camera compared to DSLR. The Z6 had some drawbacks in terms of acquiring focus on fast-moving birds and sticking with them while using the 200-500mm lens, so that was disappointing. I must add that prior to the Z6 I had never focused much on trying to photograph birds in flight, so I can't comment on whether my previous DSLR's were any better with the same lens.
When the Z6II launched I jumped on that and traded the Z6 on a new Z6II. I believe the Z6II is better at locking focus on a flying bird with the 200-500mm lens than the Z6 was. It is still not anywhere near as capable as the D500 or D850, though. I have also never used the 200-500mm lens on one of those more capable DSLR bodies, so it's very likely that the slow focus of this lens makes it less than ideal for fast action photography on ANY camera body.
There is talk from some well-informed people that Nikon is in the process of writing new firmware for the Z6/7II that will significantly improve the camera's autofocus capabilities, and that a firmware update will be released in the near future. If that happens it may well improve the ZII bodies to the point where they perform almost as well as the D500 or D850. I do believe that there is a lot of unrealized potential in the second Expeed processor that was added to the ZII bodies that could still be unlocked through improved firmware.
So for myself, now that I have more of an interest in BIF photography than before, I'm waiting for two things:
1) New, improved firmware for my Z6II.
2) Nikon's Z 200-600mm lens that is shown on the Nikon lens roadmap. I do plan to purchase that lens as soon as it becomes available.
When the above two things happen for me I'll be in a better position to evaluate whether my gear will perform well enough to satisfy my modest needs for BIF photography. I'm hopeful ...... I cannot justify spending more than 3 or 4 times as much on a better body (Z9) and a faster, much more expensive long lens. For anything other than BIF, my Z6II and my other lenses are more than good enough for the photography I do for a hobby.
So Lakeman, I agree with others here that you likely have two issues; a less capable body (one that is not likely to improve much with firmware updates), and a lens that is not best for action photography. Depending on how deeply you wish to dive into BIF photography you may have to consider replacing both camera and lens. Speaking for myself, though, I would never revert back to a DSLR at this time, no matter how much better they may currently be for BIF compared to the Z6/7(II) cameras. I just like the operation of the mirrorless bodies that much more compared to DSLR's. You spoke of a D780. That camera is probably no better with the 200-500mm lens for BIF than the Z6II, so I would not consider that body seriously if BIF is a major focus for you.
Sticking with your 200-500mm lens you may not realize enough improvement in fast action photography with a D500 or D850 body to make the switch worthwhile. On the other hand, a more capable, faster-focusing lens may also perform even better on your Z6. If I were in your position I would wait a bit longer to see whether the new Z 200-600mm lens works well for BIF, and/or whether the anticipated firmware update makes the Z6/7II significantly better for BIF. You should realize that neither of the Z6/7 cameras or the D500/D850 DSLR's are even close to the performance of the Z9 for action photography. The question is how much are you willing to spend, and how soon do you need that satisfaction?
