State functions are not exclusive concepts of Thermodynamics. As noted in the question, also in mechanics state functions appear. Potential energy has probably the closest analogy to thermodynamic state functions because it allows the evaluation of a quantity (the mechanical work) that, in general, is process-dependent (in mechanics, one would say path-dependent). However, most of the mechanical quantities are state functions, as most of them can be expressed in terms of the dynamic state variables (positions and velocities) at a given time. So, force, angular momentum, acceleration, the center of mass position, etc., are all mechanical state functions. Only in exceptional cases non-state-functions appear. For example, in materials with memory the instantaneous state is not enough to determine the force.
Due to the ubiquitous presence of state dependent quantities in mechanics, the presence of process dependent basic quantities like work or heat in thermodynamics comes as a surprise. However, it can be considered a consequence of the coarse-graining that enables us to use only a few macroscopic quantities, in place of the huge number of microscopic degrees of freedom.
Let's examine thermodynamic internal energy. Apparently, it looks like a redundant concept. After all, we know that the microscopic description of atoms, molecules, nuclei and electrons is based on Quantum Mechanics and Hamiltonian systems. No surprise that there is a total energy of the system. So, do we really need a First Principle of Thermodynamics?
The reason is that we have no way to measure the energy of a macroscopic system as a function of its microscopic degrees of freedom. Therefore, we need to introduce a concept related, but not coinciding with the total microscopic energy, only based on macroscopic measurements. We identify two process-dependent quantities (heat and thermodynamic work), that in general are not state-functions, and we discover experimentally that their sum defines a state-function. That's the reason for introducing the First Principle.
A similar approach holds for other state-functions, even those without an obvious mechanical counterpart like the entropy.
Once a useful set of state functions has been introduced in Thermodynamics, their usefulness is mainly due to the possibility of say something about a process by knowing only the initial and final state. For example, the introduction of the state-function entropy allows us to decide if a process can spontaneously happen in an system under isothermal and isobaric conditions, just by evaluating the difference of the state-function Gibbs free energy between final and initial state.
