Christofell Symbols: Derivation in Schutz & Wald

[HomogenousCow]

In Schutz, the christofell symbols are dervied from applying the product rule to a vector in a curvillinear basis.
In Wald, the christofell symbols are dervied by making an ansatz of the form a covariant derivative must take and then imposing conditions on it like the metric covariant derivative being zero.

What I find odd about this is that the derivation in Wald implies that one could construct other theories of general relativity by imposing different conditions like a non-zero torsion or a non-zero metric covariant derivative, while the derivation in Schutz leaves no room for a different theory.
Why is this so?

 

[WannabeNewton]

Well Wald's definition assumes $\nabla_{[a}\nabla_{b]}f = 0$ so you can't use the affine connections Wald defines if you want a theory of gravity with torsion. Regardless, in general we can easily have affine connections with torsion that are still metric compatible (see Wald exercise 3.1) and define a theory of gravity using this connection such as Einstein-Cartan theory. Schutz and Wald are simply excluding these cases because they are gravitational theories that are distinct from GR whereas the contents of the books only deal with GR, that's all there is to it.