For me as a mathematician, the reason why Laplacians (yes, there is a plethora of notions of Laplacians) are ubiquitous in physics is not any symmetry of space. Laplacians also appear naturally when we discuss physical field theories on geometries other than Euclidean space.
I would say, the importance of Laplacians is due to the following reasons:
(i) the potential energy of many physical systems can be modeled (up to errors of third order) by the Dirichlet energy $E(u)$ of a function $u$ that describes the state of the system.
(ii) critical points of $E$, that is functions $u$ with $DE(u) = 0$, correspond to static solutions and
(iii) the Laplacian is essentially the $L^2$-gradient of the Dirichlet energy.
To make the last statement precise, let $(M,g)$ be a compact Riemannian manifold with volume density $\mathrm{vol}$. As an example, you may think of $M \subset \mathbb{R}^3$ being a bounded domain (with sufficiently smooth boundary) and of $\mathrm{vol}$ as the standard Euclidean way of integration. Important: The domain is allowed to be nonsymmetric.
Then the Dirichlet energy of a (sufficiently differentiable) function $u \colon M \to \mathbb{R}$ is given by
$$E(u) = \frac{1}{2}\int_M \langle \mathrm{grad} (u), \mathrm{grad} (u)\rangle \, \mathrm{vol}.$$
Let $v \colon M \to \mathbb{R}$ be a further (sufficiently differentiable) function. Then the derivative of $E$ in direction of $v$ is given by
$$DE(u)\,v = \int_M \langle \mathrm{grad}(u), \mathrm{grad}(v) \rangle \, \mathrm{vol}.$$
Integration by parts leads to
$$\begin{aligned}DE(u)\,v
&= \int_{\partial M} \langle \mathrm{grad}(u), N\rangle \, v \, \mathrm{vol}_{\partial M}- \int_M \langle \mathrm{div} (\mathrm{grad}(u)), v \rangle \, \mathrm{vol}
\\
&= \int_{\partial M} \langle \mathrm{grad}(u), N \rangle \, v \, \mathrm{vol}_{\partial M}- \int_M g( \Delta u, v ) \, \mathrm{vol},
\end{aligned}$$
where $N$ denotes the unit outward normal of $M$.
Usually one has to take certain boundary conditions on $u$ into account. The so-called Dirichlet boundary conditions are easiest to discuss. Suppose we want to minimize $E(u)$ subject to $u|_{\partial M} = u_0$. Then any allowed variation (a so-called infinitesimal displacement) $v$ of $u$ has to satisfy $v_{\partial M} = 0$. That means if $u$ is a minimizer of our optimization problem, then it has to satisfy
$$ 0 = DE(u) \, v = - \int_M g( \Delta u, v ) \, \mathrm{vol} \quad \text{for all smooth $v \colon M \to \mathbb{R}$ with $v_{\partial M} = 0$.}$$
By the fundamental lemma of calculus of variations, this leads to the Poisson equation
$$ \left\{\begin{array}{rcll}
- \Delta u &= &0, &\text{in the interior of $M$,}\\
u_{\partial M} &= &u_0.
\end{array}\right.$$
Notice that this did not require the choice of any coordinates, making these entities and computations covariant in the Einsteinian sense.
This argumentation can also be generalized to more general (vector-valued, tensor-valued, spinor-valued, or whatever-you-like-valued) fields $u$.
Actually, this can also be generalized to Lorentzian manifolds $(M,g)$ (where the metric $g$ has signature $(\pm , \mp,\dotsc, \mp)$); then $E(u)$ coincides with the action of the system, critical points of $E$ correspond to dynamic solutions, and the resulting Laplacian of $g$ coincides with the wave operator (or d'Alembert operator) $\square$.