I asked this question previously to little fanfare and stumbled across it again, so I want to formulate something a bit more precise. I think it is an inherently interesting question.
Suppose that we are given a parallelizable Riemannian manifold $M$ of dimension $n$ with Riemannian metric $g_x$ at each point $x$, and likewise let $G_x$ be the matrix representation of $g_x$. We assume that all the matrices $G_x$ are relative to the same global frame of bases, so that this way $G_x$ varies smoothly on $M$ as $x$ does. Note that we could pick a global ONB and make $G_x$ constantly the identity, but this is the opposite of what we want to do.
For a given $G_x$, this matrix has a number of decompositions, particularly since it is positive definite so we can often stay inside the reals. For example, take the QR decomposition $G_x = Q_x \cdot R_x$ and consider the induced smooth functions $q, r$ from $M$ into the Lie groups $O(n)$ and $\mathbb{t}_n$ taking $x$ to $Q_x$ and $R_x$ respectively. In particular, for each smooth global frame $F$ we obtain some such maps $q_F, r_F$.
What can be discerned from properties of either a fixed pair of these maps, or the family as a whole? Can all global frames on a parallelizable Riemannian manifold be deformed into one another in a way as to induce homotopies between any two $q_F, q_H$ (up to the action of the Lie group on itself, since obv there are potential connectedness issues in some of the groups), for example? It seems plausible at least in low dimensions. Here we are conjuring large families of natural, computable maps from a manifold into Lie groups, so surely there is something interesting. The question can be asked about a variety of matrix decompositions instead of QR.
Can we compute things like the rank of the homology groups, bounds on the dimension of the isometry group, simple homotopy class, detection of spheres and handles, etc. from close inspection of these maps? It seems highly likely to me that there should be a symbolic formalism for translating statements about some such decompositions and the underlying topology/smooth structure.
I am afraid that there is a misunderstanding here: As you say in your post, you can always make your function to be constant the identity. But this exactly tells you that for a fixed parallelizable manifold $M$, any Riemannian metric $g$ on $M$ leads to the same family of smooth functions from $M$ to the space of symmetric matrices, namely all functions of the form $x\mapsto f(x)f(x)^t$, where $f:M\to GL(n,\mathbb R)$ is a smooth function. Thus you cannot learn anything about a specific metric on $M$ in that way.
Generally speaking, the problem is that global frames on a parallelizable manifold are very easy to understand, the subtle question is how they are related to holonomic frames (i.e. those coming from local coordinates). To learn something about a specific metric on $M$, the question to ask is how close the global orthonormal frame is to local holonomic frames (and this basically is what Cartan's method of moving frames does locally).
It may be possible to get information about the topology of $M$ in such a way, but as you can see from the above, the input you plan to use is rather crude.