Ahh! the old problem... equations versus reality!
All that the Einstenian equations tell us is that they don't know how to manage black holes beyond the event horizon (and that they are wrong about them because of that).
Ahh, the old problem of equations versus your imagination of what reality might be. ;)
Einstein's equations work just fine inside the event horizon. It's the actual singularity itself which raises some eyebrows. And even then, we don't actually know that such a thing isn't possible in reality. But you are uncomfortable with the idea, therefore they're wrong. Got it.
At the very least it's clear that a black hole must have "density significantly higher than that of a neutron star."
Because it's required by those pesky equations.
All you can say is that *if* (a big if) black holes behave more or less like all the physics we know about
Everything said about just about anything can be presumed to have an "as long as our understanding is correct" qualifyer. And indeed our theory of gravity may be wrong. It saying something different than what you say is not a good argument for it being wrong.
What's misleading about saying density is defined as mass against volume?
Because the actual definition is mass divided by the minimum volume which fully contains the mass, not some arbitrary larger volume that you happen to find convenient. And since the behavior of a black hole requires exceedingly high density, it's doubly misleading.