A lot of things can be derived from more basic or at least more beautiful principles. It makes perfect sense to say that Schroedinger's equation can be derived from Lagrangian Mechanics, even if this is not historically how it happened.
An especially beautiful constraint is "Schroedinger's equation has to be linear because otherwise you can build a (quantum) computer that can solve NP-complete problems in polynomial time". Schroedinger did not know about NP-completeness, but this notion does give us clues why the equation is what it is.
It is always instructive to see how a "fundamental" law can be derived or at least constrained from other laws. Thermodynamics and the notion of entropy would not have been invented if engineers did not "rederive" what was considered fundamental at the time.
The grandparent post is right in the sense that reductionism always reduces, ultimately, to a guess (in this case, Lagrangian mechanics is the guess).
You're right, though, in that a basic framework can be established, using minimal assumptions and axioms, in which most of classical physics and even much of quantum physics can be derived via (mostly) rigorous mathematical arguments. But that framework itself is, at the end of the day, a guess. It's simply the best guess we have at the given moment in time that fits the evidence (e.g., by allowing us to derive from it known rules and patterns that past generations have verified fit the evidence).
An especially beautiful constraint is "Schroedinger's equation has to be linear because otherwise you can build a (quantum) computer that can solve NP-complete problems in polynomial time". Schroedinger did not know about NP-completeness, but this notion does give us clues why the equation is what it is.
It is always instructive to see how a "fundamental" law can be derived or at least constrained from other laws. Thermodynamics and the notion of entropy would not have been invented if engineers did not "rederive" what was considered fundamental at the time.