Taylor's theorem is about regular algebraic representations $\pi$. Most representations are not algebraic in any sense (cf. my answer to this question), hence they are not connected to any Galois representation (as far as we know).

Now for an algebraic $\pi$ the question arises where to look for the corresponding Galois representation. The difficulty with $GL_2/K$ is that the corresponding symmetric space is not an algebraic variety, hence there is no cohomology which would carry the relevant Galois representation. Taylor's idea was to pass to $GSp_4/\mathbb{Q}$ which is better in that regard, but the switch requires the condition on the central character. My feeling is that what is missing is a better (more general) construction yielding an object with a Galois action, not a better understanding of the fundamental lemma or Galois representations per se. This is not my field of expertise, so I better stop here.

Taylor's theorem is about regular algebraic representations $\pi$. Most representations are not algebraic in any sense (cf. my answer to this question), hence they are not connected to any Galois representation (as far as we know).

Now for an algebraic $\pi$ the question arises where to look for the corresponding Galois representation. The difficulty with $GL_2/K$ is that the corresponding symmetric space is not an algebraic variety, hence there is no cohomology which would carry the relevant Galois representation. Taylor's idea was to pass to $GSp_4/\mathbb{Q}$ which is better in that regard, but the switch requires the condition on the central character. My feeling is that what is missing is a better (more general) construction yielding an object with a Galois action, not a better understanding of the fundamental lemma or Galois representations per se. This is not my field of expertise, so I better stop here.