There would be infinite m/v there, and finite everywhere without infinite density, so literally every single random draw of a single mass-unit would be in an area of infinite density, or so goes the idea.

That's presupposing that the mass portion of m/v is infinite, which, from our understanding of black holes, we have no reason to believe. Pathological local densities tell you nothing about the mass distribution of the universe precisely because they're so pathological.

Every theory has a domain of applicability, determined by tests within that domain. I don't think its reasonable to assume that the domain of applicability of any our theories extends to various infinities.