Maintaining symmetry? Minkowski metric is found to be
$$ds^2=-dt^2+dr^2+r^2d\Omega^2$$
where $d\Omega^2$ is the metric on a unit two-sphere.
Why should we keep track of the $d\Omega^2$ so that spherical symmetry holds well?
 A: What we mean by spherical symmetry is that if we take our geometry and consider the surface at constant $r$ it will have the same geometry as a spherical shell, that is the metric will be:
$$ ds^2 = R^2 \left( d\theta^2 + \sin^2\theta \, d\phi^2 \right) \tag{1} $$
where $R$ is some arbitrary constant. If we refer back to your previous question we find a proposal for writing the metric as:
$$ ds^2 = -e^{2\alpha(r)}dt^2 + e^{2\beta(r)}dr^2 + e^{2\gamma(r)}r^2 \left( d\theta^2 + \sin^2\theta \, d\phi^2 \right) \tag{2} $$
with $\alpha(r)$, $\beta(r)$ and $\gamma(r)$ being arbitrary functions of $r$. Taking a spherical shell means considering constant $r$ and $t$, so $dt = dr = 0$, and equation (2) becomes:
$$\begin{align}
 ds^2 &= e^{2\gamma(r)}r^2 \left( d\theta^2 + \sin^2\theta \, d\phi^2 \right) \\
      &= R^2 \left( d\theta^2 + \sin^2\theta \, d\phi^2 \right)
\end{align}$$
where the constant $R = e^{2\gamma(r)}r$. Since this is the same as equation (1) we know that it is a spherically symmetric metric.
Given the above it should be obvious that if we mess with the form of $d\Omega^2$ we won't get a spherically symmetric metric. For example we could extend our metric (2) to:
$$ ds^2 = -e^{2\alpha(r)}dt^2 + e^{2\beta(r)}dr^2 + e^{2\gamma(r)}r^2 d\theta^2 + e^{2\delta(r)}r^2 \sin^2\theta \, d\phi^2 \tag{3} $$
But at constant $t$ and $r$ we get:
$$ ds^2 = e^{2\gamma(r)}r^2 d\theta^2 + e^{2\delta(r)}r^2 \sin^2\theta \, d\phi^2 $$
and this cannot be written in the form of equation (1) so it does not have spherical symmetry.
A: See your own question here: Why do people put exponentials there
You can multiply by functions of $r$ on each of the terms.  What you cannot generally do is "muck" with the $r^2 d\Omega$ part itself.  That's what gives you the part of the distance that corresponds to the angular directions on the sphere.  If you change that, then depending on where you are, you'll get different proper distance traveled for different amounts of angle traversed.  I'm being a little sloppy in the language here, but this is the basic idea.
