The electrostatic regime only needs to consider that $\rho\neq \rho(t)$ and $\mathbf{J=0}$ and from there all the common results follow. Notice that $\mathbf{J=0}$ comes naturally if you assume that charges do not change with time, that is, that they do not move.
However the magnetostatic regime assumes $\mathbf{J\neq J}(t)$ and $\rho=0$. However, current density is the motion of charge, that is, there must be charge for there to be current density. Therefore, how is $\rho=0$ justified?
The continuity equation says that the current density is solenoidal if the charge density does not change with time ( $ \nabla \cdot {\bf J} = 0 $ ). Therefore, the field lines of $ {\bf J} $ must be closed, and the $\dot \rho = 0$ condition is compatible with the presence of currents. Think, for example, the current in a superconducting ring.
The presence of a current density different from zero, without charge density, can be easily explained by recalling that two equal and opposite fluxes of positive and negative charge may be electrically neutral while corresponding to a non-zero current.
