So I know that supermassive black holes can have "densities" less than water because black hole density scales as $1/m^2$ since $R_s = 2GM/c^2$. I am trying to reconcile this with the fact that black holes are the most compact objects for a given mass.
For instance, if I had 6 billion solar masses of material and squeezed it to the Schwarzschild radius, I would have a black hole. However, by my statement above, if I had 6 billion solar masses of water, this would occupy a volume less than the Schwarzschild radius, but would not be a black hole. How does this work? I know black holes actually have singularities (etc), but where is the flaw in my classical model of just density = M/V?