Is it not the case that F=ma and other Newtonian laws are encoded in the Lagrangian, and therefore not derived from symmetries? After all nature and her symmetries alone do not tell us how her physics works; there has to be some rule for time-evolution as well, and that's what's encoded in L (L's form is essentially a list of pairings of variables and their costs of evolution, which for classical mechanics is L = T - V = ∫ p·dv + ∫ F·dx, which says "the cost of changing v is p and the cost of changing x is F").

(QFT sort-of has an explanation for time evolution in terms of symmetries alone, but it requires a lot more machinery. But afaik classical mechanics does not.)