Definition of density of black holes

For a blackhole of mass $M$, radius $R$ and Schwarzschild radius $R^*$ (where $R<R^*$), its density $rho$ is defined as $$rho=frac{M}{(R^*)^3}.$$

One reason I read for using the Schwarzschild radius $R^*$ instead of the actual radius $R$ is that no measurements can be made inside of $R^*$ (i.e. for $r<R^*$).

I know that the Schwarzschild metric is $$g_{munu}=text{diag}left[left(-1+{R^*over r}right),left(1-{R^*over r}right)^{-1},r^2,r^2sin^2thetaright],$$
so there is a singular point at $r=R^*$ where $g_{rr}$ is infinite and hence the invariant interval $ds^2$ is infinite.

However, for points inside $R^*$ (i.e. $r<R^*$), the metric and $ds^2$ is finite. Why then is it not possible to make a measurement inside for points inside $R^*$? Is it somehow required to first pass the singular point at $r=R^*$ before a measurement can be made inside $R^*$?