I will address injectivity of $\pi_1(X_s) \to \pi_1(X)$ in the characteristic zero case. Exactness at the other places needs a proof too, but is easier (and you can find it in SGA).

I assume that the genus of $\overline{X}_s$ is $g > 0$ as the case $g = 0$ is trivial.

Let $S' \to S$ be a finite \'etale morphism with $S'$ connected. Let $X' = X \times_S S'$. Then $X'$ is connected too (by the arguments used to prove the exactness in the other places). Then $\pi_1(X') \to \pi_1(X)$ is injective, hence it suffices to prove $\pi_1(X_s) \to \pi_1(X')$ is injective.

Pick an integer $n > 2$. After a replacement as in the previous paragraph we may assume the monodromy action on $H^1_{\acute{e}tale}(\overline{X}_s, \mathbf{Z}/n\mathbf{Z})$ is trivial. We may also assume that $S$ is a scheme over $\mathbf{Q}(\zeta_n)$. Thus there is a morphism $S \to {}_n\mathcal{M}_{g, 1}$ such that $\overline{X}$ and its section are the pullback from the universal family. Here ${}_n\mathcal{M}_{g, 1}$ is the moduli stack of $1$-pointed smooth projective genus $g$ curves endowed with a "symplectic" basis of $n$-torsion on its Jacobian. By our choice of $n$ the algebraic stack ${}_n\mathcal{M}_{g, 1}$ is a smooth scheme ${}_nM_{g, 1}$. By functoriality of the fundamental group we reduce to the case $S = {}_nM_{g, 1}$ over $\mathbf{Q}(\zeta_n)$.

You can look at Petersen's answer above and insert your favorite computation of $\pi_2({}_nM_{g, 1})$ here but I want to give a more elementary proof.

In the universal case the question can be translated into a question about moduli of curves. I will do this for you without giving you all the details, but I think you will be convinced. Namely, let $G$ be a finite group. Consider the moduli stack ${}_{G, n}\mathcal{M}_{g, 1}$ parametrizing connected finite \'etale $G$-covers $D \to C$ with $C = \overline{C} \setminus \{c\}$ where $\overline{C}$ is a smooth projective genus $g$ curve endowed with a "symplectic" basis of $n$-torsion in its Jacobian. The natural map
$$
{}_{G, n}\mathcal{M}_{g, 1} \longrightarrow {}_n\mathcal{M}_{g, 1}
$$
is \'etale by deformation theory and proper by the valuative criterion along dvrs (it is not finite bc it may not be representable). To finish the proof we just need to show that ${}_{G, n}\mathcal{M}_{g, 1}$ is a stacky quotient of the form $[M'/H]$ where $M' \to {}_nM_{g, 1}$ is a finite \'etale morphism of schemes and $H$ is a finite group acting on $M'$ over ${}_nM_{g, 1}$. This is clear because we can let $M'$ be the moduli scheme parametrizing some rigidification of the problem defining ${}_{G, n}\mathcal{M}_{g, 1}$, for example add in a "symplectic" basis for the $n$-torsion on the Jacobian of $D$.

Why does this prove the result? Well, it shows that any $G$-cover of the geometric generic fibre of the universal curve spreads out over some finite \'etale cover $M'$ of the moduli space. Cheers!