I know that the Feynman diagrams and perturbation series are used as computational tools to evaluate a specified Lagrangian for a QFT. This is usually used as an argument in itself to discourage beginners in the field from "assigning reality" to the Feynman diagrams.
But I'm curious what other maybe more physically motivated arguments can be given for assigning a more fundamental tag to either the Lagrangian formulation or the reverse - a fundamental specification of Feynman diagrams (or other graph-like diagrams) and treating the Lagrangian as a sort of effective result in the limit of infinitely summed diagrams?
I know there are "non-perturbative effects" but I'm far from skilled enough to understand if they 1) are real and measurable or 2) refer only to a particular class of graph expansions, like around the coupling constant, or fundamentally to all possible breakdowns.
For example the answers in this question seem useful but are a bit too technical for me: What are non-perturbative effects and how do we handle them?
My motivation on posing the question is that even if you treat the Lagrangian as the fundamental idea, you still need to input it into the machinery with summing over all possible Lagrangians to get the probabilities of the field configurations. It would just feel nice if there was a way to break down the Lagrangian formulations in bits that fit that machinery in a more direct way (yes I know this motivation is just handwaving).
Edit: let me give a concrete toy example. I want to calculate the relative probabilities of a photon that starts at S and that can hit each of three detectors (D1-D3) after a two-slit screen.
In the Lagrangian formulation, in essence, the Lagrangian is modelled to penalize field configurations that aren't observed in nature. Wavelike propagation, the correct field<->field interactions, no discontinuities, stuff like that. But you can really input any set of field configurations. You can calculate by it by perturbation theory but you can also calculate non-perturbatively (like in lattice QCD).
So in alternative 1, I would randomly generate field configurations (for the photon field in this case and I might make a constraint that in my "universe" topology, the two slits are available and nothing else, so I don't have to model the electron field). I would filter each of these generated field configurations in 3 bins depending on if they contain a photon at Start and a photon at D1, D2 and D3. Pass the configurations through L for each bin and sum through the Path Integral to get a quantum amplitude. Normalizing and squaring and dividing gets my probabilities for S->D1, ->D2 and ->D3.
As an alternative 2, I can take the "Feynman approach" and draw a photon propagator line from S to each of the two screen holes and to each of D1-D3, assign quantum amplitudes to each path by the Feynman rules, and integrate and do the same normalization, squaring and dividing to get my relative probabilities as in the Lagrangian case.
I would assume that in this toy case I get exactly the same results. Would you have any arguments on which of these are more intuitively fundamental or useful for building further intuition on? Why exactly is the Feynman option discouraged? As we build in more advanced features in the theory, does it hit a limit in expressability?
