What does it mean if general relativity has no realistic solutions?

Effective theories

The assumed divergence of the perturbation series of QED is usually explained with a heuristic conjecture that we do not know physics at the Planck scale.

People have coined a term effective theory to describe a situation where a theory works well at some length or energy scales, but may diverge for very short distances.

General relativity

We know that Newton's gravity is very accurate for weak gravitational fields.

The linearization of the Einstein equations describes well binary pulsars, gravitational lensing, and gravitational waves.

If we try to solve the Einstein equations through some iterative approximation method, it may happen that the results are extremely accurate for a few first terms.

If the iteration anyway diverges, what might explain that? Can the divergence come from phenomena at the Planck scale?

A better explanation is that the Einstein  equations treat some quantity as absolutely rigid. The quantity cannot stretch and adapt, so that a solution could be found.

We have seen that Birkhoff's theorem means absolute rigidity with respect to the energy conservation of a spherically symmetric isolated system.

Reza Mansouri's result shows absolute rigidity with respect to an equation of state p = p(ϱ). General relativity simply refuses to comply with such an equation of state, even though the equation is reasonable and might describe a realistic physical system.

A rubber sheet model does not have such rigidity. It is intuitively clear that a rubber sheet model adapts to many types of lagrangians. There is no need for the lagrangian to conserve energy. The equation of state in Reza Mansouri's result would pose no problem for a rubber sheet.