I have three Z cameras and have sold all my F-mount cameras. I still have a bunch of F-mount long lenses including the 200-500. It's a very good lens and works very well with the FTZ on the Z cameras. While it does not work well with a teleconverter on an F-mount, it works pretty well with the 1.4 TC on the Z6/Z7 original and ii versions.
The downside of these consumer tele-zoom lenses is they are relatively slow focusing - and that impacts birds in flight, but is not that big a deal for slower moving or static subjects. That's a design decision that's part of keeping the cost down. The 200-500 as well as the 150-600 lenses in general all extend when they zoom, so the travel to move lens element groups is longer than you would ideally prefer. But in doing so with an f/5.6 or f/6.3 lens, you reduce size, cost, and have a pretty good zoom. As with some other lenses, if you pre-focus near the right distance, the issue is largely resolved.
The 100-400 is an outstanding lens - an S lens - but it's relatively expensive and not really long enough for birds. Optically it's excellent. It's a great choice for mammals and wading birds. It works with a teleconverter, but if you are going to be using it with a TC most of the time, there are other options.
The roadmap has a 200-600 lens for Z mount in the next 12-18 months. That's a good replacement targeted for the Z mount. They also have 400mm PF and 800mm PF lenses on the roadmap depending on what you need. Of course, the cost of a new Z lens will be higher than an existing F-mount lens, so there is a price for incremental performance.
I'd take a look at your budget. You can get a very good used Nikon 200-500 for $850 in a private sale, or $1100-1200 from B&H, KEH, etc. The 100-400 is $3000+ and the 200-600 will probably be in the $1400-2000+ range. The PF lenses are typically more expensive than the zooms. If you can buy the 200-500 right, or get the firmware updated for your 150-600 by Tamron, you have the luxury of waiting and taking your time. I'd probably go that route unless your budget is planning $3000+ for lens options.