The lens that is mounted to my Nikon Z9 is the Z 70-200mm f/2.8 nearly 100% of the time. I've used and owned the 100-400mm from Sigma (and used the Nikon) and the 60-600mm from Sigma along with the 24-70, and many other primes ... AND continue to keep that 70-200mm mounted. With good reason.
Before we get too far, check out some resources on how aperture and setting work with shooting action. These resources are nowhere the end all, be all -- but they are a great starting point.
The 70-200 f/2.8: Versatile and Effective
For me, I find that this lens is fast, it's reliable, and consistently hits the shot. At least that's the case for the Nikkor Z. I think it's important to point out that the Z lens is great for speed and reliability. However, if you are a hobbyist, starting out, or generally don't want to or can't spend that much on glass, don't worry, you have options. So what are the factors that might be considered?
Quality ... if you grab a lens that has a higher DXOMark score (see below), you will get amazing image quality. Even the lower-cost Sigma and Tamron lenses are OUTSTANDING
Shooting Distance ... 70mm is short enough to grab photos of the family, and 200mm is long enough to catch Addison in action
Handheld ... the 70-200mm at f/2.8 can be easily handheld and for most will not require a tripod or monopod
Teleconverter ... you can extend the 'mm' with a teleconverter and at f/2.8 as the base aperture for the lends, adding something like a Nikon Z TELECONVERTER TC-1.4x | Nikon Z Series (nikonusa.com), you can get a lens that shoots 98-280mm at f/4. The important part is the f/4 at 280mm (note: adding a teleconverter will increase the aperture)
Aperture ... the f/2.8 is a must for hiking and low-light conditions. We're frequently out when the weather turns south and the f/2.8 allows me to push shutter speeds
Sure you can get more distance with a 100-400mm, but you're also going to be shooting at f/5.6 or even f/6.3 with those lenses and that's going to push the ISO through the roof because many light conditions cal for under f/4. That is unless you're paying the $14,000 for the pro lens that's sub-f/4 and really can't be handheld. However I f that's the case, I'm jealous.
Which 70-200mm Should I Buy?
This is a great question. The Nikkor Z lens will run you upwards of $2800. It's a pro lens and is spectacular glass. All the same with the Sony G and Canon RF glass. However, there are many ways to look at lenses to determine which is the best for your particular price range. One resource is DXOMARK - Quality Testing, Scores and Reviews.
Just a few screenshots from the DXOMark website with the filter for 70-200 at f/2.8 and you'll see a price range from about $800-3000. The score is an important factor. If you're a pro, you don't need me to tell you why you should get pro-glass. Even if you're not a pro, you don't need someone telling you to go pro.
If you can afford the higher cost glass and you enjoy photography enough, it might make sense. If it's just a hobby, there are plenty of options for you. Frankly, I try to buy my glass used and that saves a lot. In the case of the Nikkor Z 70-200mm, when I purchased the glass, there weren't very many options in the used market. However, I have found plenty of Sigma and Tamron glass that performs exceptionally well.
Recap: Shooting Action (Especially with Dogs) - Go with a 70-200mm f/2.8
The 70-200mm is versatile. It's easy to carry. It's useful. With the f/2.8 aperture, there is no better bang for your buck in my opinion -- you get distance and capability. With the lower aperture, you can keep that ISO lower. Mind you, the low aperture comes at a cost with depth of field, that's a whole other issue to discuss (BLOG POST: Action Photography Series: PART 1 — Basics of Bird Dog and Upland Hunting Photography).
Get outside and point that camera at something moving! And happy hiking!