The emergence of global symmetry at low energies is a familiar phenomena, for example Baryon number emerges in the context of the standard model as "accidental" symmetry. Meaning at low energies it is approximately valid, but at high energies it is not.
The reason this is the case is that it so happens that the lowest dimension operator you can write, with the matter content and symmetries of the standard model, is dimension 5. The effect is then suppressed by one power of some high energy scale - it is an irrelevant operator. This is a model independent way to characterize the possibility of the emergence of global symmetries at low energies.
We can then ask about Lorentz invariance - what are the possible violations of Lorentz invariance at low energies, and what is the dimensions of the corresponding operators. This depends on the matter content and symmetries - for the system describing graphene, there is such emergence. For anything containing the matter content of the standard model, there are lots and lots of relevant operators*, whose effect is enhanced at low energies - meaning that Lorentz violating effects, even small ones at high energies, get magnified as opposed to suppressed at observable energies.
Of course, once we include gravity Lorentz invariance is now a gauge symmetry, which makes its violation not just phenomenologically unpleasant, but also theoretically unsound. It will lead to all the inconsistencies which necessitates the introduction of gauge freedom to start with, negative norm states and violations of unitarity etc. etc.
- At least 46, which were written down by Coleman and Glashow (Phys.Rev. D59, 116008). Relaxing their assumptions you can find even more. Each one of them would correspond to a new fine-tuning problem (like the cosmological constant problem, or the hierarchy problem).
The mechanism for "giving mass" to elementary bosons and fermions is different.
With bosons, it is related to the gauge symmetry ($SU(3)_c \times SU(2)_L \times U(1)_Y$) which is partially broken (and become $SU(3)_c \times U(1)_{em})$. The unbroken part imposes its associated bosons (gluons and photon) to be massless to respect this symmetry.
With fermions, there is no such constraint since their mass does not come from a gauge symmetry (with our current knowledge, fermions masses are put by hand via add hoc yukawa couplings). Therefore, the mass of the fermions is not predicted (contrary to the masses of bosons). So, asking "why do we see no fundamental massless fermions?", is equivalent as asking "why do we see fundamental fermions with their actual mass?". Answer: we don't know!
Best Answer
Dirac fermions is only the direct sum of left- and right-handed Weyl representations (which leads to time inversion, charge inversion and spatial inversion invariance of the theory). Two Weyl representations are mixed by the mass term in the Dirac equation. If we set mass to zero, we will get two uncoupled equations, each of which describes Weyl fermion. But the Dirac theory remains the theory of direct sum of Weyl spinors even in massless case.