We tackle the question of whether regular black holes or other alternatives to the Schwarzschild solution can arise from an action principle in quantum gravity. Focusing on an asymptotic expansion of such solutions and inspecting the corresponding field equations, we demonstrate that their realization within a principle of stationary action would require either fine-tuning, or strong infrared nonlocalities in the gravitational effective action. This points to an incompatibility between large-distance locality and many of the proposed alternatives to classical black holes.