Derivation of group velocity using Fourier transform

The aim is to determine the group velocity of a wave packet with the general form

$$\Psi\left(x,t\right)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} \phi\left(x\right)e^{i\left(kx-\omega t\right)}dk.$$

Quoting from Introducting to Quantum Mechanics by David Griffiths:

Since the integrand is negligible except in the vicinity of $\ k_{0}$, we may as well Taylor expand the function $\ \omega\left(k\right)$ about that point and keep only the leading terms: $$\omega\left(k\right) \approx w_{0}+w_{0}'\left(k-k_{0}\right)$$ where $\omega_{0}'$ is the derivative of $\omega$ with respect to $k$

What is unclear to me here is why do we Taylor-expand the function $\omega\left(k\right)$

Proceeding on, the author performed a change of coordinate variables from $k$ to $s\equiv k-k_{0}$ so that

$$\Psi\left(x,t\right)\approx \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} \phi\left(k_{0}+s\right)e^{i\left[\left(k_{0}+s\right)x-\left(\omega_{0}+\omega_{0}'s\right)t\right]}ds$$

Perhaps I am not understanding the preceding argument completely but what is the motivation for performing this change of variables from $k$ to $s$?

again, the author performed a change of coordinate so that we have

$$\Psi\left(x,t\right)\approx \frac{1}{\sqrt{2\pi}}e^{i\left(-\omega_{0}t+k_{0}\omega_{0}t\right)}\int_{-\infty}^{\infty} \phi\left(k_{0}+s\right)e^{i\left(k_{0}+s\right)\left(x-\omega_{0}'t\right)}ds.$$

A fuzzy intuition I can conjure to account for the change of variables is that the wave packet behaviours like a wave front(I.e., ablation of a layer of material that sublimates so ideally, the best way to prevent the coordinate system from 'moving' is to introduce a very similar change of variable. An explanation would really be ideal.

As the author says, he performs the change of variable "to center the integral at $k_0$". And, in turn, why? Well, you can see that he expands $\omega$ in a Taylor series around $k_0$, so he finds convenient to express the integral also centered at $k_0$. But most importantly, that change of variables is not relevant at all to reach the result the author is seeking for (you can indeed omit it and reach the same expressions without the loss of any physics), namely $$\psi(x,t)\simeq e^{-i(\omega_0-k_0\omega'_0)t} \psi(x-\omega'_0 t, 0),$$ which in turn allows to discuss the phase and group velocities.