That's not what Noether's theorem means! The "symmetries" referred to here are operations which leave invariant the action of the physical system (heuristically, the physical laws governing the system). To the extent that one can (at least in principle) write down the action of composite systems as the sum of their actions + interactions, these are taken to mean (again heuristically) the laws of physics writ large. Likewise, "conservation law" here means not specifically the particular result of an experiment (like the work integral you describe), but a more general notion of conservation, in the sense of there being universally invariant conserved quantities (i.e. things that cannot be created or destroyed).

To get a feel for it (and why it may not necessarily be intuitive) consider the following pairs of symmetries and conservation laws:

- The laws of physics are invariant under time translation (i.e. repeating an experiment at different times gives you the same result ceteris paribus). The corresponding conserved quantity is energy.

- The laws of physics are invariant under spatial translation (i.e. repeating an experiment at two different places gives you the same result ceteris paribus). The corresponding conserved quantity is momentum (this is a vector: one component of momentum for each possible direction of translation).

- The laws of physics are invariant under rotation (i.e. repeating an experiment under different orientations gives you the same result ceteris paribus). The corresponding conserved quantity is angular momentum (again a vector, since the rotation group SO(3) has three generators).

- Electromagnetism, constructed as a classical field, exhibits an internal symmetry in the field quantities. The corresponding conserved quantity is the electrostatic charge (actually a 4-vector current).

It is not a priori obvious from from the kind of arguments you've supplied why these should be the case. Demonstrating these would require access to the Lagrangian formalism (i.e. action principles) and how it behaves under these symmetry operations. I would say that, as you suspect, you are fundamentally misunderstanding the situation.