The older 70-200 VR usually referred to as the VR1 only after the VR2 came out is build well, is fast and in the center rather sharp but its weakness was soft corners in Fx but no problem with the smaller image circle of Dx so to save money a used VR1 is a great buy, since they were so well made they will still be in good shape and save you $1500-2000. On the D500, its field of view is the same as a 300mm so it has enough reach for most wildlife unless you are after small birds or insects 50 meters away. It vibration reduction is good so a lot of shots can be made without a tripod and relatively slow shutter if the light is low.
A new alternative is the 70-300 AFP which has a fast pulse motor and very good optics compared to all prior versions of the 70-300. For not as much weight or size as expected for a 300mm lens, the Nikon 300mm E f4 pf ed vr is a miracle of tech with Fresnel lens so its normally very heavy objective lens is a fraction of the weight and length of any 300 f/4....and sharper. An east to carry small very sharp fast 300 that does not need a tripod means it is unique in the industry. There is a 500mm version, that on a Dx body has the same field of view as a 750mm lens is out now but demand is far outstripping supply so they are hard to find in some markets but in Europe they seem to be instock in many stores. I rented the 300FP and on my D850 and later on the Z6, it performed beautifully and weighed less than my 70-200 and was shorter
I do not shoot wildlife much or sports but a 70-200 2.8 is my most used lens, that and 24-70.2.8 of f/4 out of 15 Nikkor fast lenses and a few Sigma and a very nice Tamron 15-30 2.8 G2. As a result of very high satisfaction with the Tamron G2, I would also have no hesitation in recommending that also and it being 1/2 the price of a Nikon, if you want new with warranty, check it out.
Regarding the body, in DX, which has advantages in wildlife and sports, there is no DSLR on the planet that is a competitor to the D500. Canon has their 7DII but it never came close and rumor is that it is not being replaced with a newer model, Canon effectively conceding the Dx sport and Wildlife market to Nikon. The only other competing camera is the Sony a9 which has very good AF but is not built anywhere near what outdoorsmen expect of their rugged reliable weather resistant wildlands camera. It also suffered from the worst support and repair facility record in the industry.
A big plus of the D500 is also the same thing that many reviewers(who are pushing other cameras for which they get ample revenue) is used of the only pro card system out and the ability to use future releases of the CF Express cards with simply a firmware update. XQD is orders of magnitude more reliable than consumer based SD card system.
For those with a more all round photography style, the Fx Z6 is better in most uses, with better low light IQ, better AF in low light, better VF by far, smaller, lighter and access to the low cost and industry topping S lenses. For video hybrid shooters, it is a no brainer with the best video of any FX cameras, and any F mount lens investment is protected by using the FTZ adaptor.