Having botched the first attempt at answering this question and not wanting to delete the evidence, let me try again here.

The "tetrad postulate" is independent from metricity and from the condition that the connection be torsion-free. It is simply the equivalence (via the vielbein) of two connections on two different bundles. Here are the details. $M$ is a smooth $n$-dimensional manifold.

First of all we have an affine connection $\nabla$ on $TM$ with connection coefficients $\Gamma^\rho_{\mu\nu}$ relative to a coordinate basis -- that is, $$\nabla_{\partial_\mu} \partial_\nu = \Gamma_{\mu\nu}^\rho \partial_\rho,$$ with $\partial_\mu$ an abbreviation for $\partial/\partial x^\mu$ where $x^\mu$ is a local chart on $M$.

Then we have a connection on an associated vector bundle to the frame bundle $P_{\mathrm{GL}}(M)$. The frame bundle is a principal $\mathrm{GL}(n)$-bundle and given any representation $\rho: \mathrm{GL}(n) \to \mathrm{GL}(V)$ of $\mathrm{GL}(n)$ we can define a vector bundle $$P_{\mathrm{GL}}(M) \times_\rho V.$$ Take $V$ to be the defining $n$-dimensional representation and call the resulting bundle $E$.

A vielbein then gives a bundle isomorphism $TM \to E$ and the "tetrad postulate" is telling you how to define a connection on $E$ from the affine connection $\nabla$ on $TM$. In effect, it gives you an expression for $\omega$ in terms of the vielbeins and $\Gamma$.

This works for any affine connection $\nabla$ on any smooth manifold $M$. No metric is involved.

A special case of this construction is when $(M,g)$ is a riemannian manifold and $\nabla$ is the Levi-Civita connection (i.e., the unique torsion-free, metric connection on $TM$). You can without loss of generality restrict to orthonormal frames, which defines a principal $\mathrm{O}(n)$ (or $\mathrm{O}(p,q)$ depending on signature) bundle. The representation $V$ restricts to an irreducible rep of the orthogonal group, possessing an invariant bilinear form $\eta$. This relates $g$ and $\eta$ as in your question.

Having botched the first attempt at answering this question and not wanting to delete the evidence, let me try again here.

The "tetrad postulate" is independent from metricity and from the condition that the connection be torsion-free. It is simply the equivalence (via the vielbein) of two connections on two different bundles. Here are the details. $M$ is a smooth $n$-dimensional manifold.

First of all we have an affine connection $\nabla$ on $TM$ with connection coefficients $\Gamma^\rho_{\mu\nu}$ relative to a coordinate basis -- that is, $$\nabla_{\partial_\mu} \partial_\nu = \Gamma_{\mu\nu}^\rho \partial_\rho,$$ with $\partial_\mu$ an abbreviation for $\partial/\partial x^\mu$ where $x^\mu$ is a local chart on $M$.

Then we have a connection on an associated vector bundle to the frame bundle $P_{\mathrm{GL}}(M)$. The frame bundle is a principal $\mathrm{GL}(n)$-bundle and given any representation $\rho: \mathrm{GL}(n) \to \mathrm{GL}(V)$ of $\mathrm{GL}(n)$ we can define a vector bundle $$P_{\mathrm{GL}}(M) \times_\rho V.$$ Take $V$ to be the defining $n$-dimensional representation and call the resulting bundle $E$. Relative to a local frame $e_a$ for $E$, a connection $\hat\nabla$ defines connection one-form $\omega$ by $$\hat\nabla_{\partial_\mu} e_a = \omega_{\mu~a}^b e_b.$$

Now the vielbein defines a bundle isomorphism $TM \buildrel\cong\over\longrightarrow E$ and all the "tetrad postulate" says is that the two connections $\nabla$ and $\hat\nabla$ correspond. In fact, the "tetrad postulate" is just the statement that the vielbein is a parallel section of the bundle $T^*M \otimes E$ relative to the tensor product connection.

This works for any affine connection $\nabla$ on any smooth manifold $M$. No metric is involved.

A special case of this construction is when $(M,g)$ is a riemannian manifold and $\nabla$ is the Levi-Civita connection (i.e., the unique torsion-free, metric connection on $TM$). You can without loss of generality restrict to orthonormal frames, which defines a principal $\mathrm{O}(n)$ (or $\mathrm{O}(p,q)$ depending on signature) bundle. The representation $V$ restricts to an irreducible rep of the orthogonal group, possessing an invariant bilinear form $\eta$. This relates $g$ and $\eta$ as in your question.