I have owned Sony A6500, A7RII, A7RIII, A7III, A9, and currently own A7RIV, A1, A7SIII.

My recommendation would be to get the Sony A7IV, it has all the latest Sony processors and all the best of Sony Autofocus prowess with less cost than an A1, plus a fully articulating screen, full touch menu and excellent video capabilities and so much more including Human, Animal and Bird Eye-AF.

The 200-600mm lens is super. I have owned one since they first came out in August 2019 and I have traveled the world with it making great shots with my A9, A7RIV, and my A1, and also using both Sony 1.4X and 2X TCs with it. .

First a shot with the 200-600mm lens and my 24mp A9. Kids being pulled by a speedboat on Brookings lake, Manistee National Forest, Michigan, USA 24mp Sony A9, 200-600mm with 1.4X TC, 840mm

Second and third shots, with the 200-600mm lens and my 61mp A7RIV, a Muscovy Duck starts its takeoff from a pond on the Caribbean Island of Sint Maarten/Sint Martin. First the full frame, then a tight crop of the same shot. 600mm shot.

Fourth shot, a Snowy Egret takes off from its watery perch on Sint Maarten/St. Martin. Sony 50mp A1, 200-600mm lens , 391 mm, 1/4000 sec.

Fifth and sixth shots, A Great White Egret bends its neck to clean its feathers on Sint Maarten/St. Martin. A1, 200-600mm lens, 600mm. First the full frame, then a tight crop from same image.

Seventh shot, two wild iguanas getting cozy together, Island of Sint Maarten/St. martin. Sony A1, 200-600mm lens, 300mm

Eighth shot, at a wild Elephant Seal rookery along the Pacific Ocean in California, a young pup squeals as it tries not to get crushed to death by a 5000 pound amorous Bull going after one of its cows. This time this pup survived. Sony A7RIV, 200-600mm lens, 600mm

Ninth and tenth shot, the same kids (as in first shot) a couple of years older being pulled on Brookings Lake, Manistee National Forest, Michigan. A1, 200-600mm lens with 1.4X TC, 840mm, 1/2000 sec.

Bird Eye-AF, Human Eye-AF and Animal Eye-AF used in these shots, and Yes the A7IV has all these AF modes.

Cheers and best to you all.

After reading everyone's prior comments (and based on the following constraints) I am closer to choosing a mirrorless camera and lens.

What's important
Less than $6000.
New and not used since I have had terrible experience with used cars!
Same brand lens as camera.
Consider a lens and camera that I can extend to reach further with compatible teleconverter. (a later on possibility)
Consider a better lens. Unfortunately I can't afford a prime 600 mm and probably wouldnt be able to lift it anyway.
Consider highest mpixels possible since I crop like crazy.
No heavier than my current Tamron 250-600 and D800 since I shoot without a tripod.
Must be good for catching birds in flight from far away. Birds in flight are the primary use.

Not so important
Switching to another system (My brain can still handle that ..... I think)
Noise at very high iso's is not so important since I usually don't run too high an iso
I really don't care if my image is soft at the edges since I almost always crop the edges off and I will use my D800 for landscapes and other photos

So right now I am pretty sure I will use a Sony FE 200- 600 f/5.6-6.3 GOSS.
I am leaning towards a Sony 33 mpixel A7 IV but am also considering a Sony 61 mpixel A7R IV
the A7 has AI that picks up on birds, the A7R does not.
the A7 of course has half of the mpixels as A7R but I wonder if it is really needed if I have a better lens and supposedly better auto focusing and tracking with the A7.
I would appreciate any comments from anyone familiar with this equipment and especially its use on birds in flight.
I really appreciate the help I was given before and I am getting closer, I think to what I want.

Ordered and just received a refurbished Z6II, but the battery was missing. When I notified Nikon of the missing part, the representative had me complete a form with the camera's serial number and description of the missing part to be returned. Thought I would get an acknowledgment pretty quickly that they were sending the battery but nothing yet, so I called back to be sure the claim form was received...and it was. I asked if the 14 days inspection time would be re-set based on when the battery was received and was told NO. The battery may take a few days to process, plus shipping time. That severely cuts down on my test period since a few days have already elapsed.

So, just a heads up that Nikon does not re-set the 14-day window for returning equipment even if a part was missing from the order they sent.