Recently I attended a very short course on string theory. We went through the standard presentation in light-cone gauge for brevity. We ‘derived’ the Einstein field equation in the following manner. Partial gauge fixing in the conformal gauge leaves a residual (local) conformal symmetry. Since gauge (local) symmetries must be preserved at the quantum level for self-consistency, the beta function (which generically breaks scale and thus conformal symmetry by introducing anomalous dimensions c.f. Callan-Symanzik equation) must be zero (after fixing the dimension to be critical).
The claim is that $\beta=0$ gives you the Einstein field equations. On the other hand, $\beta=0$ is not usually considered as a dynamical equation. E.g. the Standard Model is a chiral gauge theory whose (gauge) chiral anomaly is zero due to cancellation from the charges of the fields. The lecturer told me some dodgy answer about how the induced effective theory had the Einstein field equations as dynamical equations, but I’m interested in considering string theory as a fundamental theory. The effective theory is irrelevant.
It seems to me we are merely pruning the configuration space of dynamically allowed fields to ensure self-consistency. I.e. string theory is not self-consistent: we need something extra to impose the Einstein field equations.
Why can we set $\beta =0$ when this is never done in any other field theory?