The Sony 200-600 is unquestionably the best lens for sports/wildlife/birds so all that remains is the BODY and AF......
By rights, you should be shooting with a crop frame body - but the A6600 is NOT ideal - HOPEFULLY Sony is working on a sports/wildlife crop body ! A friend shoots with the A7RIV and the A9. The A9 has better fast/accurate AF and is what he uses for wildlife - but only 24 MP for cropping. That leaves the A1 as the optimal body and blows your budget !
Remember, the minimum is 600mm FF equiv. and the ability to get to 840mm with GOOD quality......
.

Nikon Z9.... $5500
Sony A1..... $6500

Not sure which part of "less than $6000" people don't understand. With either of those models that folks have recommended repeatedly you would use all your budget or blow completely through it.

I really don't have a recommendation. Just a general observation.

Personally when I shoot birds in flight I tend to do so with shorter telephotos.... 300mm on APS-C, 400mm on full frame are pretty much my maximums. Longer than that can be difficult to keep the narrow angle of view on fast moving subjects.

Might want to look at the EXIF of examples shown here and your own photos, to see what focal lengths you and others are actually using for BIF.

(Hint: Of the actual BIF photo examples others have shared here, two were shot at 391mm and one was at 105mm, all on full frame. Can't tell how much cropping, if any, was done to any of those images. Personally I try to avoid cropping as much as possible. But of course it's not always possible.)

Do you need 600mm or even more? Maybe yes for perched birds, but good luck keeping long focal lengths/narrow angle of view on fast moving subjects!

Since you referenced my two BIF shots i have posted, just let you know both my shots are nearly fullframe, little to no cropping. In numerous wildlife, long lens and other facebook photography groups there are loads of BIF shots done at 600mm without TCs, or even more with TCs. The A1 , A7IV and modern E-mount lenses have no problems locking focus, tracking and shooting BIF at longer than 319mm. LOL

And the action shots I have posted of shots made at 840mm also should answer your questions.

Take a look at the new Wildlife Pro Mark Smith video I linked. Mark labels the still shots and you can see the focal length of the BIF shots.

Here just a quick grab shot at great distance with some atmospheric haze here, as I spun and fired with my Sony A7RIV and Sony 200-600mm lens at 600mm, full frame width with just a crop down from the open sky at the top of the frame, of a flying Pelican off the Caribbean Island of Sint Maarten/St. Martin, framed by the silhouette of the island of Saba. In an instant the older model A7RIV which has excellent AF, but not quite that of the latest AF of the A1 and A7IV, still easily grabbed focus on the Pelican and held it, and didn't move over to focus on the island in the distance.

I have 48 years of practice as a pro in following moving subjects with long lenses in all shooting situations, so it isn't any hardship for me to track BIF at long focal lengths. Neither is it hard for anyone with practice and the right gear.

A note: The Island of Saba may look familiar, it was used as the original Kong Island, in the original King Kong movie, a fact.

Cheers and best to you all.

Since you referenced my two BIF shots i have posted, just let you know both my shots are nearly fullframe, little to no cropping. In numerous wildlife, long lens and other facebook photography groups there are loads of BIF shots done at 600mm without TCs, or even more with TCs. The A1 , A7IV and modern E-mount lenses have no problems locking focus, tracking and shooting BIF at longer than 319mm. LOL

And the action shots I have posted of shots made at 840mm also should answer your questions.

Take a look at the new Wildlife Pro Mark Smith video I linked. Mark labels the still shots and you can see the focal length of the BIF shots.

Here just a quick grab shot at great distance with some atmospheric haze here, as I spun and fired with my Sony A7RIV and Sony 200-600mm lens at 600mm, full frame width with just a crop down from the open sky at the top of the frame, of a flying Pelican off the Caribbean Island of Sint Maarten/St. Martin, framed by the silhouette of the island of Saba. In an instant the older model A7RIV which has excellent AF, but not quite that of the latest AF of the A1 and A7IV, still easily grabbed focus on the Pelican and held it, and didn't move over to focus on the island in the distance.

I have 48 years of practice as a pro in following moving subjects with long lenses in all shooting situations, so it isn't any hardship for me to track BIF at long focal lengths. Neither is it hard for anyone with practice and the right gear.

A note: The Island of Saba may look familiar, it was used as the original Kong Island, in the original King Kong movie, a fact.

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.

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.