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The three-dimensional Laplacian can be defined as $nabla^2=fracpartial^2partial x^2+fracpartial^2partial y^2+fracpartial^2partial z^2$. Expressed in spherical coordinates, it does not have such a nice form. But I could define a different operator (let's call it a "Laspherian") which would simply be the following:

$$bigcirc^2=fracpartial^2partial rho^2+fracpartial^2partial theta^2+fracpartial^2partial phi^2$$

This looks nice in spherical coordinates, but if I tried to express the Laspherian in Cartesian coordinates, it would be messier.

Mathematically, both operators seem perfectly valid to me. But there are so many equations in physics that use the Laplacian, yet none that use the Laspherian. So why does nature like Cartesian coordinates so much better?

Or has my understanding of this gone totally wrong?

Nature appears to be rotationally symmetric, favoring no particular direction. The Laplacian is the only second-order differential operator obeying this property. Your "Laspherian" instead depends on the choice of polar axis used to define the spherical coordinates.

Now, at first glance the Laplacian seems to depend on the choice of $x$, $y$, and $z$ axes, but it actually doesn't. To see this, consider switching to a different set of axes, with associated coordinates $x'$, $y'$, and $z'$. If they are related by
$$mathbfx = R mathbfx'$$
where $R$ is a rotation matrix, then the derivative with respect to $mathbfx'$ is, by the chain rule,
$$fracpartialpartial mathbfx' = fracpartial mathbfxpartial mathbfx' fracpartialpartial mathbfx = R fracpartialpartial mathbfx.$$
The Laplacian in the primed coordinates is
$$nabla'^2 = left( fracpartialpartial mathbfx' right) cdot left( fracpartialpartial mathbfx' right) = left(R fracpartialpartial mathbfx right) cdot left(R fracpartialpartial mathbfx right) = fracpartialpartial mathbfx cdot (R^T R) fracpartialpartial mathbfx = left( fracpartialpartial mathbfx right) cdot left( fracpartialpartial mathbfx right)$$
since $R^T R = I$ for rotation matrices, and hence is equal to the Laplacian in the original Cartesian coordinates.

To make the rotational symmetry more manifest, you could alternatively define the Laplacian of a function $f$ in terms of the deviation of that function $f$ from the average value of $f$ on a small sphere centered around each point. That is, the Laplacian measures concavity in a rotationally invariant way. This is derived in an elegant coordinate-free manner here.

The Laplacian looks nice in Cartesian coordinates because the coordinate axes are straight and orthogonal, and hence measure volumes straightforwardly: the volume element is $dV = dx dy dz$ without any extra factors. This can be seen from the general expression for the Laplacian,
$$nabla^2 f = frac1sqrtg partial_ileft(sqrtg, partial^i fright)$$
where $g$ is the determinant of the metric tensor. The Laplacian only takes the simple form $partial_i partial^i f$ when $g$ is constant.

Given all this, you might still wonder why the Laplacian is so common. It's simply because there are so few ways to write down partial differential equations that are low-order in time derivatives (required by Newton's second law, or at a deeper level, because Lagrangian mechanics is otherwise pathological), low-order in spatial derivatives, linear, translationally invariant, time invariant, and rotationally symmetric. There are essentially only five possibilities: the heat/diffusion, wave, Laplace, Schrodinger, and Klein-Gordan equations.

The paucity of options leads one to imagine an "underlying unity" of nature, which Feynman explains in similar terms:

Is it possible that this is the clue? That the thing which is common to all the phenomena is the space, the framework into which the physics is put? As long as things are reasonably smooth in space, then the important things that will be involved will be the rates of change of quantities with position in space. That is why we always get an equation with a gradient. The derivatives must appear in the form of a gradient or a divergence; because the laws of physics are independent of direction, they must be expressible in vector form. The equations of electrostatics are the simplest vector equations that one can get which involve only the spatial derivatives of quantities. Any other simple problem—or simplification of a complicated problem—must look like electrostatics. What is common to all our problems is that they involve space and that we have imitated what is actually a complicated phenomenon by a simple differential equation.

At a deeper level, the reason for the linearity and the low-order spatial derivatives is that in both cases, higher-order terms will generically become less important at long distances. This reasoning is radically generalized by the Wilsonian renormalization group, one of the most important tools in physics today. Using it, one can show that even rotational symmetry can emerge from a non-rotationally symmetric underlying space, such as a crystal lattice. One can even use it to argue the uniqueness of entire theories, as done by Feynman for electromagnetism.