Comment by xigency
It's evident and obvious in any of these explanations that the equations and properties of math are taken as true a priori, not grounded on observation (in their invocation).
If I write a partial differential equation that I came up with randomly and ask you to find all the potential solutions that really doesn't tell you anything about the natural world.
I think that's more your interpretation/experience rather than the intention of the tools. Those constants and coefficients are there because the math is describing the shape of the solution based on logic, mathematical object rules, and symmetry/conservation laws and needs to be "grounded" to make them physical.
The Lagrangian is just "conservation of energy" (L = T[kinetic] - V[potential]). There isn't some magic, it's a statement that the energy needs to go somewhere.
Your straw-man belies the underlying issue you are experiencing, you don't just come up with a PDE, you see nature and then you describe ways to conserve counts of things, "energy", "population", whatever. The PDEs describe the exchange between these counts. The accuracy and additional terms are about more accurately representing the counts and conservation of things.