We investigate the resistively shunted Josephson junction (RSJ) at equilibrium, using linear response, an exact path integral technique and symmetry considerations. All three approaches independently lead to conclude that the superconducting - insulating quantum phase transition long believed to occur in the RSJ, cannot exist. For all parameters, we find that shunting a junction makes it more superconducting. We reveal that the UV cutoff of the resistor plays an unforeseen key role in these systems, and show that the erroneous prediction of an insulating state resulted in part from assuming it would not. We also explain why the RSJ physics differs from that of 1D quantum impurity problems. Our results fully support and confirm the experimental invalidation of this quantum phase transition by Murani et al. in 2020.