A particle dropping into a Schwarzschild black hole takes an infinite time to reach the horizon if observed from the outside. Thus, if a Schwarzschild black hole forming in a collapse is observed, this would suggest that, at any finite time, some mass at least is located outside the horizon, near to it.
On the other hand, if you look at the pure Schwarzschild solution, this is based on an energy-momentum tensor which vanishes everywhere, in particular at the horizon.
The singularity theorems show that solutions to the Einstein field equations may have singularities under generic conditions. The relevance of this is that singularities are not artifacts which appear only under very special symmetry conditions (which conditions might be considered without physical relevance in real astrophysical situations).