Why are physicists consistently considering models that are breaking the symmetries of the theory they are expressed in? 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).
 A: You have an extremely restrictive opinion on what should be called physics.  I suspect you would find very few people to agree that one is only doing physics if one imposes full Galilean or Lorentz symmetry on all of their models; that would rule out almost everything in vast swaths of the physics community.

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?

If you are referring to the Kepler problem with $\mathbf F \propto -\frac{\mathbf r}{r^3}$, then it possesses rotational and temporal symmetry but not translational symmetry, and so angular momentum and energy are conserved while linear momentum is not.
If you prefer, you can consider the dynamics of two masses $m_1$ and $m_2$, under the influence of an attractive force with magnitude $F \propto \frac{\mathbf r_1 - \mathbf r_2}{|\mathbf r_1-\mathbf r_2|^3}$.  This system possesses full Galilean symmetry, and conserves linear momentum, angular momentum, and energy.

It seems to me that we cannot even know when it i the case, so why are we even using them to begin with?

Conservation laws hold if the corresponding symmetries hold, as per Noether's theorem.  We know which conservation laws to expect because we know which symmetries apply.

Isn't this kind of practice extremely bad and confusing for students too?

No.
A: I'm not really sure I understand your question, but I'll try to answer anyway.
You start with classical mechanics. Here, I guess the main points are:

*

*Classical mechanics itself (i.e. the "physical laws") are invariant under translations in space and time and under rotations, hence energy, momentum and angular momentum are conserved.

*You can consider any system of masses with forces between them, and the conservation still holds. In other words, the physical system does not have to be invariant under the transformations. (Otherwise, physics would be pretty boring.)

*You often encounter systems which can be, to a very good approximation, described in much simpler terms by taking some limit, e.g. taking the sun as infinitely heavy for the Kepler problem, taking the earth as infinitely heavy and flat for stuff moving on the surface, taking one end of a spring to be fixed to a rigid point (which is again fixed to the earth) etc. In all these cases, you end up with a set of physical laws that have different (usually fewer) symmetries than the orginal, and thus less conserved quantities (ball bouncing off the earth etc.). In all of these cases, the procedure is not that hard, and you can go through the details to see what the symmetries are, and, of course, for which limits the approximation holds.

*In the end, you have to do the same thing for QFT, and that might be technically and conceptually  more challenging. You might e.g. prefer dimensional to cutoff regularisation because Lorentz invariance is manifest, but in the end you have to carefully check that your procedure doesn't introduce some strange behaviour. Sometimes that's easy, sometimes it's hard -- an obvious case would be anomalies in a quantised theory, which took some time to understand.
One complication is that an unrenormalised theory in itself is ill-defined in that e.g. amplitudes are given by divergent integrals. In that sense, the renormalised theory is the real one, while the starting point is more of a heuristic motivation. On the other hand, field content and symmetries (IF your renormalisation procedure can maintain them) are carried over.

