I think that the apparent puzzle originates, at least in part, from our conceptual difficulty to abandon the macroscopic-world identification between momentum and velocity.
In classical mechanics, velocity is a derived concept, justified and rooted on the existence of a continuous trajectory, something which is well defined at the macroscopic scale. In the quantum description there is nothing allowing to justify the idea of a smooth trajectory. Thus, no velocity, at least according to the classical mechanics definition. Instead, in QM, one can still introduce a measurable quantity, the momentum of a particle, for example through a quantitative analysis of scattering experiments.
Taking into account the previous remark, let's analyze the situation of a hydrogen atom with its electron in the 1s state.
On the one hand, no doubt that the local probability current must be zero everywhere, since the 1s state can always be described by a real function. In physical terms this is related to the absence of an electric current in the 1s state.
On the other hand, a measurement of momentum may give any value according to a distribution probability provided by the squared modulus of the 3D Fourier transform of the 1s wave-function.
The two facts are not in contradiction: the vanishing of the local current is telling that in average, taking an ensemble of H atoms all in their 1s state, there is no electron displacement or circulation in a well defined direction. However, in a single measurement, nothing forbids the possibility of finding any value for the momentum.