I tend to think of the Levi-Civita connection as a unique connection on a given Riemannian manifold that allows it to "mimic" being an embedded submanifold of $\mathbb{R}^n$. Considering the historical development of differential geometry, which most certainly began by considering curves and surfaces in $\mathbb{R}^3$, many of the nice properties of surfaces embedded in a real space would have naturally been desirable in more general cases divorced from the assumption of a Euclidean ambient space. In order to see what I mean let's consider an arbitrary coordinate system, $x^i$, on $\mathbb{R}^n$ as well as some position vector;
$$\vec{R}(x^i)$$
which emanates from the origin of our coordinate system. Now, we can construct a basis for $\mathbb{R}^n$ by simply taking the partial derivative of this position vector with respect to the $x^i$;
$$e_j(x^i):=\frac{\partial \vec{R}(x^i)}{\partial x^j}$$
Notice that, depending on the structure of our chosen coordinate system, the elements of our basis may change from point to point in $\mathbb{R}^n$. Therefore, taking the derivative of our basis elements then yields;
$$\frac{\partial e_j}{\partial x^k}=\frac{\partial^2 \vec{R}}{\partial x^j \partial x^k}:=\Gamma^i_{jk}e_i$$
Here we see that the torsion free property of the Levi-Civita connection can be interpreted as a consequence of the fact that partial derivatives commute rather than as a purely arbitrary imposition on our connection coefficients. $\\$ Now, constructing the elements of our metric tensor, $g$, by taking pairwise dot products of our basis vectors yields;
$$g_{ij}=e_i\cdot e_j=g_{ji}$$
Then taking the partial derivative of the metric tensor coefficients yields;
$$\frac{\partial g_{ij}}{\partial x^m}=\frac{\partial}{\partial x^m}(e_i\cdot e_j)=\frac{\partial e_i}{\partial x^m} \cdot e_j + e_i \cdot \frac{\partial e_j}{\partial x^m}$$
$$=\Gamma^k_{im}g_{kj}+\Gamma^k_{jm}g_{ik}$$
Similarly;
$$\frac{\partial g_{mj}}{\partial x^i}=\Gamma^k_{im}g_{kj}+\Gamma^k_{ij}g_{mk}$$
$$\frac{\partial g_{im}}{\partial x^j}=\Gamma^k_{ij}g_{km}+\Gamma^k_{jm}g_{ik}$$
Adding and subtracting these derivatives in a clever way then yields a very natural formula for obtaining the connection coefficients without any direct reference to a position vector;
$$\Gamma^k_{im}=g^{kj} \frac{1}{2}\bigg[\frac{\partial g_{ij}}{\partial x^m}+\frac{\partial g_{mj}}{\partial x^i}-\frac{\partial g_{im}}{\partial x^j}\bigg]$$
A similar analysis also yields the vanishing derivative of the metric tensor but at this point it should be intuitively clear where these seemingly arbitrary impositions are coming from. This is a point that many have made before me; Mathematics tends to tell stories in reverse historical order and set what were once viewed as consequences of some analysis as axioms. This is often very enlightening and generalizes certain results beautifully, but the cost is often obfuscation of things which were once more intuitively clear. Once we begin entering the land of intrinsic, Riemannian geometry where we completely abandon any consideration of an ambient Euclidean space we must "artificially" choose to impose certain restrictions on our connection if we want our analysis to align with our everyday experience of curves and surfaces.
