Physics theories are always constrained by symmetry principles (Strong equivalence principle, Galilean invariance principle, Gauge invariance, …).
This means that each model of a theory inherits (by construction) the symmetries enforced by the theory. For example, in classical mechanics, every model is (supposedly) Galilean invariant (no absolute frame of reference, invariance by rotations, translations, boosts, …).
Now, take the classical expression of the Lorentz force: it depends explicitly on v and and cannot therefore be Galilean invariant. The same is true for the harmonic oscillator $F = -kx$ or the central force $F = -\frac{1}{r^2}$. In these cases, an absolute frame of reference is defined because an absolute point is discriminated (the origin) from all others.
As these models are not Galilean invariant, we should not expect them to preserve energy, momentum, angular momentum or any other quantity related to these broken symmetries (say, by Noether theorem).
How to explain that in some case the conservation laws are preserved while they a priori have reason to be, say, for the central force? It seems to me that we cannot even know when it i the case, so why are we even using them to begin with?
Isn't this kind of practice extremely bad and confusing for students too?
As an example, notice that in the central force case the momentum is not preserved because you don't have invariance by translation in space as a point is discriminated and fixed.
EDIT: apparently my question wasn't clear, so here's a new formulation. Given a model $M$, some physics claims are sometimes preserved by doing a limit process or by renormalizing some quantity in $M$, and sometimes they are not. Why? How can we know which claims are preserved a priori and which are not?
Knowing what symmetry is preserved or not is just one type of physics claim (you have infinite many others you can make), and it's only trivial in models that admit a lagrangian formulation and continuous symmetries by Noether theorem. It's not even trivial otherwise (say, in stat mech models).