But it's also very obvious that general relativity can only be an effective theory that fails for very strong fields. No one I know actually thinks there are si

But it's also very obvious that general relativity can only be an effective theory that fails for very strong fields. No one I know actually thinks there are singularities, those just point to places where GR starts to significantly deviate from reality. Navier–Stokes equations are very accurate on macroscopic scale, but regardless of their mathematical properties, there are no actual blow ups in real world, because ultimately, the liquid is composed from atoms. Likewise, the fields in GR must be ultimately quantized in some way, which will almost certainly break the singularities. The infinities in QFT are really just a hack. I mean, all those calculations are perturbational in the first place, so they can be hardly considered a "true" picture of reality.