How to make a non-shaking First-person Shooter camera with subsequent adaptation to Third-person shooter?

Question

Unity

The author with a meager vocabulary, please understand and forgive. There is not normative vocabulary because I like it so much.
Dying Light

First of all, I would like to draw your attention to the fact that here the modders did not bother with adapting the animation for Third-person, since the developers did it for them. (presumably this was made for co-op)

In my opinion, the first person camera in Dying Light is simply divine, I have never seen such a responsive character in games, so I can only praise the programmers for being excellent.

Mirror's edge
Here are screenshots from a game similar to the genre:

Here, as we can see, a character model is also added, although during the passage, we see only Faith's legs and arms.

Reality
I tried to do something similar to Unity:
https://www.youtube.com/watch?v=pEc_HTRnrt8
(video with sound)
mde) I'm not much of a programmer.

Alpha version
Before that, I made another prototype of the same camera, but with a significant difference: now adding the camera to the model, and not vice versa. And I made the movement of the model itself dependent on the rotation of the camera through the Two Bone IK Constraint script (this is from the Animation Rigging Package).
https://www.youtube.com/watch?v=6F8raVZXfjs
Here, in general, the result turned out to be cool, but the problem is that the camera depends on the model, that is, if I add a walking animation, the camera will frantically fuck in all directions, in other words shake.

Okay, I thought, but then I came up with the idea of ​​creating a phantom character body. That is, I copied and pasted the same model into the FPScontroller, and now I added the camera there, but now in this model I will not turn on the walking animation, thus the camera will not fuck up.

But now there was a problem, now the camera began to lag behind the model. I haven't been able to solve this problem yet.

I used Unity 2019.3.2f1, from package I used the same Animation Rigging and Animator. I used a model from Mixamo, animations from the same place.

If you are a unity guru, please advise different resources, videos or articles on how to do this.