The mean density of the interior is much much much lower than you think, because the Schwarzschild interior volume is enormously larger than that of a closed Euclidean 3-ball of the Schwarzschild radius.

Some details:

Christodoulou & Rovelli 2014 showed that the maximal interior volume of a Schwarzschild black hole increases with time <https://arxiv.org/abs/1411.2854>. There is plenty of follow-on work in the literature for other varieties of black hole, and it is pretty generic that a black hole gets REALLY BIG inside as it gets older.

I have previously aimed HNers at DiNunno & Matzner 2008 <https://arxiv.org/abs/0801.1734> (this is also good teaching material for thinking about different systems of coordinates on a Schwarzschild BH spacetime).

One might get some intuition by thinking about pouring material into a (non-static) black hole. When do you saturate the black hole? Does it ever fill completely up? How does its outer boundary (the horizon) evolve as you pour more and more material in?