Yes, as shown by Oppenheimer and Snyder gravitational collapse takes place in finite proper time for infalling matter, but infinite coordinate time as measured by an exterior observer, such as ourselves. From our point of view, the singularity never actually forms. A "black hole" is really a "frozen star". On the other hand, infalling matter does hit the singularity in its own proper time.
The black holes studied by Hawking et al are mathematical idealisations $-$ they refer to a static solution of Einstein's equation which assumes that all matter has already formed into a singularity. Although this is a valid solution of an equation, General relativity does not provide any mechanism according to which such solutions actually exist in our universe. Consequently, one should see Hawking radiation as a mathematical exercise rather than as a physical description of matter. Whether Hawking radiation can actually apply outside of a frozen star seems to me far too complex a problem for a realistic solution.
To go further we need to have a mathematical model for what actually happens in the physics of a black hole. The meaning of the word "singularity" is that it is a point where the equations of physics no longer apply and we cannot describe the laws of physics. I doubt this applies just to the singularity itself - if the structure of spacetime breaks down, I think one would expect it to break down in a neighbourhood, not just at a point.
Quantum mechanics also describes situations in which the conventional structure of spacetime breaks down, so that we can no longer describe particles as having precise position. I believe that any deeper investigation of the structure of black holes awaits a true theory of quantum gravity. I have given more discussion in my books and in Mathematical Implications of Relationism.