What's more distressing than the insular knowledge cults of modern physics is the bizarre fixation on unfalsifiable philosophical interpretation.
That just makes it incomprehensible to outsiders when they quibble over the metaphors used to explain the equations that are used to guess what may happen experimentally. (Rather than admitting that any definition is an abstraction and any analogies or metaphors are merely pedagogical tools.)
My kneejerk reaction: Give me the equations. If they are too complicated give me a computer simulation that runs the equations. Now tell me what your experiment is and show me how to plug the numbers so that I may validate the theory.
If I wanted to have people wage war over my mind concerning what I should believe without evidence, I would turn back to religion rather than science.
Anyway, I hope this situation improves in the future. Maybe some virtual particle will appear that better mediates this field (physics).
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.
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.
Math is all you've got to work with, we wouldn't have modern day physics without math.
The issue is that people think they can find some kind of magic shortcut by playing around with abstractions without reference to or grounding in physical observables. That's not a math problem, that's a psychology problem.
Once again, my point is that people are trying to take shortcuts with abstractions that are not grounded in reality. That is a matter of self-discipline, of priorities, of putting the cart before the horse. Consider string theories: we have worked out so many ways in which strings can behave, etc. with so many possibilities and permutations. However, we never proved the ground reality for strings, we just ran with a bunch of assumptions and then parameterized them, went meta a bunch of times, and called that a research program.
All of that mathematical sophistication and model-building could have went to, e.g. perfecting QCD, or even in other directions.