I think "symmetry" is actually not a great way to think about F=ma. The reason is that Newton's laws still work even if your system is non-symmetric with non-conserved dynamical quantities.

Consider releasing a rock and calculating how long it takes to hit the ground. In this system, the rock's momentum increases in time, and is therefore not conserved.

Consider a charged particle in an oscillating electric field. In this system, the particle oscillates, and so energy is not conserved.

What about Noether's theorem? In the first example the Lagrangian has a height dependence, and in the second one there's a time dependence. So the theorem does not apply.

Yet F=ma still works. So F=ma does not come from symmetry.

You might object that these quantities are conserved globally. But the point is that F=ma allows us to use model systems as affected by external forces; this is a very powerful simplifying technique.