I don't see why the idea of steady currents (i.e. magnetostatics) implies that charge density $\rho(\vec{r},t)$ has no explicit time dependence.
Is it just coming from magnetostatics being defined as $ \vec{\nabla} \cdot \vec{J}:= 0$ (I don't see why this would be true either) and because of the continuity equation $\implies \frac{\partial\rho}{\partial t} = 0$.
Introduction to Electrodynamics, D.J. Griffiths section 5.2.1 - Steady Currents
Magnetostatics is, in some sense, a toy concept taught to students in preparation for the formal magneto-quasi-static (MQS) approximation. The purpose of the MQS approximation is to decouple the electrical field from the magnetic field. This is done by setting $\frac{\partial}{\partial t} \vec E \approx 0$ so that Ampere's law becomes $\nabla \times \vec H \approx \vec J$ (see http://web.mit.edu/6.013_book/www/chapter3/3.2.html )
Since we want $\frac{\partial}{\partial t} \vec E \approx 0$ and also $\epsilon_0 \nabla \cdot \vec E = \rho$ then that implies that $\frac{\partial}{\partial t} \rho \approx 0$
By decoupling the fields it becomes much easier to solve. So this approximation is very useful to make. Allowing $\frac{\partial}{\partial t}\rho \ne 0$ would result in $\frac{\partial}{\partial t} \vec E \ne 0$ and thus the fields would be coupled again. So this assumption actually turns out to be more important that the steady current assumption $\frac{\partial}{\partial t} \vec J \approx 0$ assumption. In fact, in the MQS the latter assumption is not made and the currents are allowed to change over time, but the equations remain decoupled and simple to solve at each time point.
I would personally define a “steady current” as one that obeys $\frac\partial{\partial t} \mathbf J = 0$ everywhere.
The requirement that the charge density in “magnetostatics” not change with time, $\frac\partial{\partial t} \rho = 0$, allows the student to use the tools developed during electrostatics to figure out what the electric fields do.
Most elementary treatments (including Griffiths) begin with a chapter or two of electrostatics with $\frac{\partial}{\partial t}\rho =0$ and $\mathbf J = 0$. Next follows a chapter or two of magnetostatics with $\mathbf J\neq0$ but $\mathbf J$ and $\rho$ both held constant. Then the charging-or-discharging capacitor is introduced as an example where there are regions of $\frac\partial{\partial t}\rho \neq 0$ and therefore $\frac\partial{\partial t}\mathbf E \neq 0$, motivating Maxwell’s discovery of the need for the displacement current. This pedagogical strategy mirrors the historical timeline of the development of the theory.
