Using auto ISO may result in a too high ISO, if you set it for that.
I use aperture priority for the purpose of using a high shutter speed. I like to shoot, if possible, 1/4000 sec. this gets the wing tips frozen.
I control the iso so my shutter speed is fast enough. If lighting conditions change, my shutter speed will change, but NOT THAT MUCH. Probably two or three stops, which is no big deal.
Here is the thing RahulKhosla, we now live in a digital world, not a film one, we can now experiment and see our results right away, so start shooting using different approaches. What works for someone else will not work for you.
So try manual with auto iso. Try aperture priority using manual iso, and letting the exposure compensation make minor adjustments. But experiment, do not take any ONE way as the only way to do something.
I use Manual, Shutter Priority, Auto ISO, Manual ISO, it all depends on what I am trying to do for the final image.
It is not about one approach, it is about the right approach under the current conditions that you are shooting under.
Black Crowned Night Heron, Nikon D500, Nikon 200-500 lens, 1/320 sec. f5.6, ISO 1600, A Priority, -1 exposure compensation.

I have been following Steve Perry's advice on Manual mode + auto ISO so far but am curious about how Aperture priority would work for situations where the subject is moving quickly (eg BIF or a tiger chasing a deer) and so you need pretty high shutter speeds to avoid motion blur (say 1/2000 - 1/4000).

So many wildlife photographers swear by Aperture priority mode and I can certainly see the logic in terms of having to fiddle with one less variable (shutter speed which the camera controls).

However, if I need reasonable DoF AND and I also simultaneously need high shutter speed, then a higher F stop to provide more DoF would automatically slow down the shutter speed, wouldn't it? And thus create the very motion blur that I want to avoid?

Cranking up the ISO or EV could be a solution but would love to hear advice on this.

Firstly, I am not aware how many use Aperture priority for wildlife or suggesting it is a good method but here's a couple of thoughts;

a) it could be that they have assessed the present lighting conditions and determined (by test or knowledge) that say for any aperture change they make between f/2.8 and f/8 that they will use to achieve the DoFs they want, the camera will give speeds in an acceptable range for the ISO they have set.

b) it could be that they use Aperture priority with Auto ISO and have a 'minimum' speed set that will cater for any animal movement expected.

I shoot with Nikon bodies. I set Aperture Priority and Auto ISO. In Auto ISO, I set the range of acceptable ISOs--usually 100 to 25600. The second setting in Auto ISO is the minimum shutter speed. And that varies for what I am shooting and the lens I am using. It could be from 1/250 to 1/3200.

I set the aperture I want by turning the command wheel on the camera. The aperture chosen again varies by conditions present and what I am shooting and backgrounds and depth of field considerations. The camera then picks the ISO needed for light present. In my nature shooting, the light is often changing and I do not want to change the ISO manually.