Well, the thing about black hole singularity is that it is, in a sense "veiled" behind the event horizon. Light can pass in, but not out, beyond the event horizon. There is no "naked" black hole singularity without an event horizon. To get really picky about it, the Schwarzschild-radius-style event horizon is only one possible solution, there are more complicated ones, but they all have in common that the singularity cannot be observed directly but is "hidden" inside the horizon. To get even more picky about it, a black hole rotating quickly enough would not have an event horizon and would be directly observable - but there is no physical process (that we have any indication of existing) which would enable a black hole to rotate at such a rate.
In any case, an observer crossing the event horizon would not "see" or "reach" the singularity either: it would always be in the future, to use a very loose manner of speaking.
To sum it up, you are right that contemporary physics has no way of describing the conditions at infinite density, which would prevail at the location of a singularity. Quantum Mechanics for example can't be applied if there is no "room" to "stow" the wavelength of a particle - and a singularity is point-like.
All that said, my days in natural science academia are long past. Please take all I wrote with a grain of salt.