Signals and disturbances are more often than not characterized in the frequency domain. Therefore, the frequency domain approach is often more powerful for robustness and disturbance analysis.
Some might also claim it helps with spelling ;-)
The expression $\hat G(s) = [\hat G_1; \hat G_2]$ is a transfer function matrix, which is generalization of SISO transfer function to MIMO. For example when both $\hat G_1$ and $\hat G_2$ are SISO, such as
$$
\hat G_1(s) = \frac{1}{s}, \quad \hat G_2(s) = \frac{1}{s+1}
$$
you get
$$
\hat G(s) =
\begin{bmatrix}
\frac{1}{s} \\ \frac{1}{s+1}
\end{bmatrix}
$$
which represents a system with one input and two outputs.
You can also stack state space models which is equivalent to $\hat G(s) = [\hat G_1; \hat G_2]$ by defining the state vector as $x=\begin{bmatrix}x_1^\top & x_2^\top\end{bmatrix}^\top\!\!\in\mathbb{R}^{n_1+n_2}$ with the following state space model
$$
A = \begin{bmatrix}A_1 & 0 \\ 0 & A_2\end{bmatrix}, \quad
B = \begin{bmatrix}B_1 \\ B_2\end{bmatrix}, \quad
C = \begin{bmatrix}C_1 & 0 \\ 0 & C_2\end{bmatrix}, \quad
D = \begin{bmatrix}D_1 \\ D_2\end{bmatrix}.
$$
If instead you want $\hat G(s) = [\hat G_1, \hat G_2]$ (so for SISO $\hat G_1$ and $\hat G_2$ you would have that $\hat G$ has two inputs and one output) you can use
$$
A = \begin{bmatrix}A_1 & 0 \\ 0 & A_2\end{bmatrix}, \quad
B = \begin{bmatrix}B_1 & 0 \\ 0 & B_2\end{bmatrix}, \quad
C = \begin{bmatrix}C_1 & C_2\end{bmatrix}, \quad
D = \begin{bmatrix}D_1 & D_2\end{bmatrix}.
$$
However this method of combining state space models might not yield minimal models, so they might either be uncontrollable or unobservable. This will for example be the case when $\hat G_1 = \hat G_2$.
Best Answer
Unlike Fourier transform
*, Laplace transforms bears very little practical or physical insight. Actually the only reason you learn partial fraction expansion in the undergrad courses is because of Laplace Transforms.The original insight of taking the Laplace transforms was to solve differential equations much more efficiently and in a more humane manner. But this got stuck and we are using it extremely blindly. 80% of control engineering students learn it as replace the number of dots with powers of s. It is very ill-taught and frankly you need to know lots of mathematics to start this type of queries if you want to do justice. For example, you stop worrying about the domain of convergence, analytic continuation and many many interesting features of this transform just after you learn about the definitions. If we are interested in stable systems and if
sis not defined on the left half plane how is it that we discuss negative real part poles etc. See it immediately gets confusing because of the terrible motivation of such concepts. It is the same problem with Dirac's delta function. It only makes sense under the integral sine but we keep on multiplying with time functions etc. as if it is a real function. Hence, things become pretty tricky if you are not exposed to these concepts.Another shortcoming of such thinking is that it forces control engineers think in terms of artificial causalities that bears no value in the original system. Take a clamped mass-spring-damper system, the equations that govern the motion is:
$$ m\ddot x(t) + b\dot x(t) + kx(t) = F(t) $$ now, if you follow the typical route, you get $$ \frac{X(s)}{F(s)} = \frac{1}{ms^2+bs+k} $$
here you must think that because of force we obtained a position output and hence the causality is position due to force. But obviously there is no such thing, the position and the force satisfy the same differential equation simultaneously. There is no such thing as force comes in and creates displacement.
My suggestion is that you keep this transform as a tool for solving/modeling differential equations as systems. Also, laplace transforms (or other integral kernels) are a much richer elements of mathematics. Hence, one should not overload them with artificial physical meanings.
*Fourier Transforms at least give some (and emphasis on some towards little) understanding of the steady state(!) behavior when the input is assumed to be pure sine. They are by no means indicative if you have a problematic transient regime.