The point is that such a theory can be constructed in a quite simple way.
First, destroy relativistic symmetry. Add some term to the Lagrangian which destroys covariance and prefers some coordinates. Then you already have a GR variant as a field theory on R4\mathbb{R}^4. GR in harmonic gauge would be the most plausible candidate.
Then, use a lattice regularization. With a small enough lattice distance on a big enough cube this will not lead to any problem with GR in our observable universe. Don't care about renormalization, use the lattice regularization as a theory itself. It is a well-defined theory with a finite number of degrees of freedom.
Then use canonical quantization to quantize it. The result is a well-defined theory of quantum gravity compatible with all existing empirical evidence. Which is anathema because it destroys, by construction, relativistic symmetry. No chance to publish it.