We apply the new quantization scheme outlined in M. Becker and M. Reuter, Phys. Rev. D 102, 125001 (2020), to explore the influence which quantum vacuum fluctuations of the spacetime metric exert on the universes of quantum Einstein gravity, which is regarded an effective theory here. The scheme promotes the principle of background independence to the level of the regularized precursors of a quantum field theory (“approximants”) and severely constrains admissible regularization schemes. Without any tuning of parameters, we find that the zero point oscillations of linear gravitons on maximally symmetric spacetimes do not create the commonly expected cosmological constant problem of a cutoff-size curvature. On the contrary, metric fluctuations are found to reduce positive curvatures to arbitrarily tiny and ultimately vanishing values when the cutoff is lifted. This suggests that flat space could be the distinguished ground state of pure quantum gravity. Our results contradict traditional beliefs founded upon background- dependent calculations whose validity must be called into question therefore.