If the chosen set of axioms does not contain Schrodinger/ Newton equations, then the formalisim should provide ways for DERIVING them from the axioms. Any failure means that your axioms contain a piece of garbage.
Noether's theorem is a THEOREM! "given A, then B"; where A is the invariance under some Lie group, and B is a statement about a conserved quantity that can be DERIVED SYSTEMATICALLY from the action integral. If this is not a derivation, can you tell us what is it that we do when we play with equations and produce B from A?
I do, because I lose my job if I cann't derive anything!
I am a physicist and I only care about a self-consistent set of axioms, I leave experiments for the more able peopel who work on them!