Many of the papers (e.g., this) dealing with nonlinear electrodynamics treat a theory's prediction of vacuum birefringence as undesirable, but don't explain why it would be undesirable. For example:

Starting with the most general non-linear theory derived from an arbitrary Lagrangian depending on two Lorentz invariants of Maxwell’s tensor, L(P,S), he discovered that among all such non-linear theories, the Born-Infeld electrodynamics is the only one ensuring the absence of birefringence, i.e. propagation along a single light-cone, and the absence of shock waves. In this respect the Born-Infeld theory is unique (except for another singular and unphysical Lagrangian L=P/S). A beautiful discussion of these properties can be found in I. Bialynicki-Birula’s paper

The bold italicized emphasis is mine.

Other papers (e.g., this) treat vacuum birefringence in extreme fields as expected and natural.

My question: is vacuum birefringence expected and natural, and Born-Infeld electrodynamics therefore deficient?

Many of the papers (e.g., this) dealing with nonlinear electrodynamics treat a theory's prediction of vacuum birefringence as undesirable, but don't explain why it would be undesirable. For example:

Starting with the most general non-linear theory derived from an arbitrary Lagrangian depending on two Lorentz invariants of Maxwell’s tensor, L(P,S), he discovered that among all such non-linear theories, the Born-Infeld electrodynamics is the only one ensuring the absence of birefringence, i.e. propagation along a single light-cone, and the absence of shock waves. In this respect the Born-Infeld theory is unique (except for another singular and unphysical Lagrangian L=P/S). A beautiful discussion of these properties can be found in I. Bialynicki-Birula’s paper

The bold italicized emphasis is mine.

Other papers (e.g., this) treat vacuum birefringence in extreme fields as expected and natural.

My question: is vacuum birefringence expected and natural, and Born-Infeld electrodynamics therefore deficient?