= Solution
Use units $c=1$, and let $r$ be the areal radial coordinate of the <FRW metric>, rather than the radial <comoving distance> when $k\ne0$. On a constant-<cosmic time> slice the spatial <metric tensor> is
$$
d\ell^2=a(t)^2\left(\frac{dr^2}{1-kr^2}+r^2d\Omega^2\right),
\qquad
dV=a^3\frac{r^2\sin\theta}{\sqrt{1-kr^2}}\,dr\,d\theta\,d\phi.
$$
The <determinant> of the spatial <metric tensor> supplies the factor $(1-kr^2)^{-1/2}$. Integrating over the <solid angle> gives $dV_{\rm shell}=4\pi a^3r^2|dr|/\sqrt{1-kr^2}$. An incoming radial <null geodesic> obeys
$$
\left|\frac{dr}{dt}\right|=\frac{\sqrt{1-kr^2}}{a(t)}.
$$
Consequently the proper shell volume sampled by an emission-time interval of positive width $dt$ is $4\pi a^2r^2dt$. Multiplying by the proper <number density> proves the <source counts on an FLRW past light cone>:
$$
\boxed{dN=4\pi a(t)^2r(t)^2n(t)\,dt.}
$$
Here $r(t)$ is evaluated on the observer's <past light cone>, not along a fixed source worldline. The curvature factor cancels between the spatial volume and the radial light speed; it has not been set to one.
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