Some of the newer 70-200 F2.8 lenses are quite excellent, even when used wide open. I also have an 80-200 F2.8 AF-D two ring design, which has been "good enough" for what I use it for. Tamron, Nikon's FL and Sigma's Sport all outperform the AF-D. Sharper, faster and more accurate focus, and a pure delight to work with. I have brief experience with the Nikkor and it blew me away. The Sigma is quite good - I borrowed a friend's copy and felt it was worth the cost. I have no experience with the Tamron.

Being fussy will cost ya . . .

For birds and wildlife a 70-200 without an extender is too short, and with a 1.4x could be used if there were no other options. I have been using a 600mmF4 for birds, and have recently acquired a Sigma Sport 150-600 and found the image quality at 600 very similar to what I was getting with the Nikkor prime, but I did have to kick up the ISO in crappy light. But still, being able to leave the tripod/gimbal head home and shoot hand held was a big plus for me. However, if I were to do it over, I would likely get the Tamron G2 version of that focal length - which is 2 lbs ligher and only costs around $1300 these days. If you want a lens that will do what you need for wildlife, I'd consider it. The image quality is good enough for "fussy" and the focal length will not leave you wishing you had a longer lens.

Generally speaking, the main advantage of F2.8 and faster lenses is not so much that you would use the lens at that aperture but more that once you reduced the aperture to something smaller, for greater depth of field perhaps, the lenses lose much of their optical flaws and weaknesses, and you still have an F4 or F5.6 lens. If you start at F4, you may not see optimum performance until F8. This is speaking very generally - individual lenses can be and are different.

Not sure how far along you are in your selection process...or how much you have read and studied yet. But if you are looking at a D500, be aware that the sensor in that camera puts almost exactly the same demands on a lens as does the sensor in the D850, and except for overall image dimensions, gives almost identical results. To get the best results requires the best lens you can put on the camera. I have not used any of the lenses you mention except for the 70-200. It sounds like there may be a fundamental conflict between your stated fussiness versus not being able to justify the price of the 70-200. That being the case, I'd suggest that you rent a couple of your possible choices and see if you can tell the difference between (or among) them. I don't believe that anything that anyone can tell you is going to help make the choice that you will be happy with.