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).