The Nikkor AF-S 70-200 VR II is not really "long enough" for a lot of wildlife photography, but since you have a 200-500mm, you have that covered.
HOWEVER, I find it a little surprising that you don't already have a 70-200mm in your kit. It's one of the most useful of all zooms. In fact, I use mine enough that I bought a second one as a backup (and, incidentally, I actually now use the backup more than my original f/2.8... the backup is a smaller/ligher f/4 version that also has great image quality).
It's also not a macro lens, only able to do 0.12X magnification on it's own (the newer AF-S 70-200 VR "FL" which superseded the "II" is closer focusing, can do 0.21X on it's own).
HOWEVER, if you get a set of macro extension tubes, you can use the 70-200 for close-ups, too. For the image below, I used a 20 or 25mm extension tube on my Canon 70-200L IS lens (which is limited in it's close-up ability, similar to the Nikkor II)....
I recommend the Kenko macro extension tube set. It's a good value at around $130 for a set of three, high quality tubes (12mm, 20mm & 36mm) which fully support all lens functions and can be used singly or stacked to provide higher magnification. I started using macro extension tubes 25+ years ago and have found them so useful I always have some in my bag, just in case they're needed. They don't take up much room or add much weight.
https://www.bhphotovideo.com/c/product/375238-REG/Kenko_AEXTUBEDGN_Auto_Extension_Tube_Set.htmlThe Nikkor AF-S 70-200mm f/2.8 VR "II" is an excellent lens... but you might want consider the even better "FL" with fluorite added, which is even better (less chromatic aberration and even sharper). The "FL" is considerably more expensive, even though it's currently on sale with a hefty discount. Also handle the f/2.8 lenses to see if you're okay with the size and weight. If you find it a bit much, the f/4 version might be a good alternative.
And, no, in my opinion, you don't need another "real" macro lens like the Nikkor 105mm. It sounds as if the Tamron 90mm is meeting your needs very well.