This form of Maxwell's Equations should cover metals dielectrics and most other media. Focus on the following
$$
\begin{align}
\boldsymbol{\nabla}.\mathbf{D}&=\rho_f \\
\boldsymbol{\nabla}\times\mathbf{H}&=\mathbf{J}_f+\partial_t\mathbf{D}
\end{align}
$$
Now take divergence of the second equation, and bear in mind that divergence of a curl is zero:
$$
\boldsymbol{\nabla}.\boldsymbol{\nabla}\times\mathbf{H}=\boldsymbol{\nabla}.\mathbf{J}_f+\boldsymbol{\nabla}.\partial_t \mathbf{D}=0
$$
You can then swap and substitute $\boldsymbol{\nabla}.\partial_t \mathbf{D}=\partial_t \boldsymbol{\nabla}.\mathbf{D}=\partial_t \rho_f$
So the continuity equation is a direct consequence of Maxwell's equations. Verifying it is akin to verifying Maxwell's equations. No need to mess around with thing wires and loops, the above prescription applies to most media and even mixed domains, i.e. with different media.
The above analysis tells you that you cannot talk about charge density and current density separately.
I am not sure what proof you are expecting for infinitely thin wire. There I would simply state current density as:
$$
\mathbf{J}\left(\mathbf{r}\right)=\alpha\int ds'\: \boldsymbol{\mathcal{\dot{R}}}\left(s'\right)\delta^{\left(3\right)}\left(\mathbf{r}-\boldsymbol{\mathcal{R}}\left(s'\right)\right)
$$
Where $\boldsymbol{\mathcal{R}}\left(s\right)$ is the trajectory of your wire, parametrized by arc-length $s$. Assuming $\boldsymbol{\mathcal{\dot{R}}}.\boldsymbol{\mathcal{\dot{R}}}=\mathcal{\dot{R}}^2=const$
Value of $\alpha$ follows from:
$$
\int_S{d^2}r\: \mathbf{\hat{n}}.\mathbf{J}=I\Delta\left(s\in S\right)\cos\theta_\mathcal{\dot{R}}=\alpha\mathcal{\dot{R}}^2\,\Delta\left(s\in S\right)
$$
Where the integral above is over the surface area $S$ of the dot-product of the current density with the surface normal $\mathcal{\hat{n}}$. Quantity $\Delta\left(s\in S\right)=1$ is the wire only goes through the surface area $S$ once. $\cos\theta_{\mathcal{\dot{R}}}$ is the cosine of the angle between the wire and the surface normal. $I$ is the current in the wire. Equation above only makes sense if wire goes through $S$ once or not at all.
From this you can then extract the charge density by taking divergence. I think, you will find that once you take divergence the derivative (of the divergence) and the integral ($\int ds'$) will cancel out, and you will get zero charge density for closed-loop wires
ADDENDUM
Lets see how certain integrals/derivatives will transform
Let us restrict our attention to a region, $\left(x,y,z\right)\in\Omega$ and $s\in\left(s_0,s_1\right)$, where $z$ has a one-to-one relationship with arc-length $s$. More specifically, $\mathcal{R}_z\left(s\right)$ gives the z-coordinate of the arc at all points. Within the region we are considering, let $\mathcal{R}_z^{-1}\left(z\right)$ be defined.
Irrespective of where $z$ is the component of the arc-tangent (of the wire) in the direction of $z$ is:
$$
\frac{d\mathcal{R}_z}{ds}=\mathbf{\hat{z}}.\frac{d\boldsymbol{\mathcal{R}}}{ds}=\mathbf{\hat{z}}.\boldsymbol{\dot{\mathcal{R}}}
$$
Next note that within $\Omega$ for any function $f=f\left(x,y,z\right)$:
$$
\begin{align}
\int^b_a dz\, f\left(x,y,z\right)&=\int^{\mathcal{R}_z^{-1}\left(b\right)}_{\mathcal{R}_z^{-1}\left(a\right)} \frac{dz}{ds} ds f\left(x,y,\mathcal{R}_z\left(s\right)\right)=\int^{\mathcal{R}_z^{-1}\left(b\right)}_{\mathcal{R}_z^{-1}\left(a\right)} \frac{d\mathcal{R}_z}{ds} ds f\left(x,y,\mathcal{R}_z\left(s\right)\right)\\
&=\mathbf{\hat{z}}.\int^{\mathcal{R}_z^{-1}\left(b\right)}_{\mathcal{R}_z^{-1}\left(a\right)} \boldsymbol{\mathcal{\dot{R}}} f\left(x,y,\mathcal{R}_z\left(s\right)\right)ds
\end{align}
$$
Where I simply replaced the integration variable $z\to s$ where $z=\mathcal{R}_z\left(s\right)$. The same trick will work in volume integrals within $\Omega$. The the transform would be:
$$
\begin{align}
x &\to x \\
y&\to y \\
z &\to s \\
\end{align}
$$
With the Jacobian $\left|\frac{\partial\left(x,y,z\right)}{\partial\left(x,y,s\right)}\right|=\mathbf{\hat{z}}.\boldsymbol{\mathcal{\dot{R}}}$
It then follows that (within $\Omega$):
$$
\begin{align}
\mathbf{\hat{z}}.\mathbf{J}\left(\mathbf{r}\right)&=\alpha\mathbf{\hat{z}}.\int ds'\: \boldsymbol{\mathcal{\dot{R}}}\left(s'\right)\delta^{\left(3\right)}\left(\mathbf{r}-\boldsymbol{\mathcal{R}}\left(s'\right)\right)=\alpha\int dz'\: \delta^{\left(3\right)}\left(\left(\begin{array}\\x\\y\\z\end{array}\right)-\left(\begin{array}\\\mathcal{R}_x\left(\mathcal{R}_z^{-1}\left(z'\right)\right)\\\mathcal{R}_y\left(\mathcal{R}_z^{-1}\left(z'\right)\right)\\z'\end{array}\right)\right)=\\
&=\alpha \:\delta\left(x-\mathcal{R}_x\left(\mathcal{R}_z^{-1}\left(z\right)\right)\right)\:\delta\left(y-\mathcal{R}_y\left(\mathcal{R}_z^{-1}\left(z\right)\right)\right)
\end{align}
$$
From here it should be relatively easy to derive:
$$
\int_{x_0}^{x_1} dx \int_{y_0}^{y_1} dy \int_{-l/2}^{l/2} dz\: \mathbf{\hat{z}}.\mathbf{J}\left(\mathbf{r}\right)=l\,\alpha
$$
Assuming that $\left(x_0,\,x_1\right)\times\left(y_0,y_1\right)\times\left(-l/2,l/2\right)$ contain the wire and are within $\Omega$. I think this is what you were after.
Another interesting thing to try is:
$$
\begin{align}
\boldsymbol{\nabla}.\mathbf{J}&=\int^{s_b}_{s_a} ds'\: \boldsymbol{\mathcal{\dot{R}}}\left(s'\right).\boldsymbol{\nabla}\delta^{\left(3\right)}\left(\mathbf{r}-\boldsymbol{\mathcal{R}}\left(s'\right)\right)=\int^{s_b}_{s_a} ds'\: \frac{d\boldsymbol{\mathcal{R}}}{ds'}.\boldsymbol{\nabla}\delta^{\left(3\right)}\left(\mathbf{r}-\boldsymbol{\mathcal{R}}\left(s'\right)\right)
\\
&=-\int^{s_b}_{s_a} ds'\: \frac{d}{ds'}.\delta^{\left(3\right)}\left(\mathbf{r}-\boldsymbol{\mathcal{R}}\left(s'\right)\right)=\delta^{\left(3\right)}\left(\mathbf{r}-\boldsymbol{\mathcal{R}}\left(s_a\right)\right)-\delta^{\left(3\right)}\left(\mathbf{r}-\boldsymbol{\mathcal{R}}\left(s_b\right)\right)
\end{align}
$$
Clearly, if $s_a=s_b$ as would be in the case of a closed loop, the divergence of the current density would vanish.
Best Answer
The presence of a current does mean charge is moving around, but not necessarily that the charge density is changing anywhere. If $\nabla \cdot \vec{j} \neq \vec{0}$, then we'd have sinks and/or sources. Since it is zero, all we have is charge flowing around with more charge coming in to replace it.
Imagine, for example, there is an infinite line of people walking at a constant speed. Of course, everyone is changing places all the time. However, the density of people is always the same, for whenever someone walks forward, the person behind fills in the space that would be left open. It is the very same idea. While charges are moving around, there is always some more charge to fill in the places that would be left open, and charges are always leaving a spot where another charge is entering. Hence, charge does move around with the current, but no sources or sinks ever occur.
A similar analogy is to observe the flow of water on a bathtub, for example. While water can move around, it has no sinks nor sources.