How stellar black holes are formed? This is a kind of follow-up of my previous Phys.SE question Can something (again) ever fall through the event horizon?  that seems to have established that, for an outside observer, things never fall into the black hole. Note that this is not to say that an outside observer never sees anything falling into a black hole (as in receiving the optical signal), but it is instead an stronger statement: no matter how long an outside observer wait for anything to fall into a black hole, there will always exist intersection between his own future light cone and the in-falling object's light cone outside the event horizon. That is what I mean by "things never fall into the black hole".
Assume an initial universe with no black holes (stellar or primordials), and thus, has no event horizons. Is there a solution to GR equations showing that the light cone of any amount of mass will inevitably reach a singularity, never intersecting with the light cone of an observer outside the would-be-black-hole? For my understanding, the Schwazschild metric describes an eternal, static, black hole, so I guess I am looking for something else here.
Please notice that I tried to be much more specific than the other questions about black hole formation, because the answers saying that in falling observers would reach the singularity in a finite amount of proper time isn't really helping me to understand the phenomenon from an outside perspective.
 A: I can't claim I have an answer but I will aim to clarify some points.
If you have no black holes to begin with, it is not necessary that you form a black hole. Therefore, a quick answer is no - if there are no singularities to begin with and no singularities are subsequently formed, then obviously nothing will ever hit a singularity.
However, if you take Schwarzschild and put some mass at rest in Schwarzshild, then they will definitely fall into the black hole and hit the singularity. As you pointed out, an outside observer wouldn't see that. Personally, I think it would be very weird just seeing an object disappearing. The picture of "things never fall into the black hole" is much more natural. 
You might be looking for a naked singularity (without an event horizon). A naked singularity is believed not to exist generically, and is related to the Weak Cosmic Censorship Conjecture. You can certainly come up mathematically with spacetimes that contain naked singularities. In those cases, it might be true that things can just disappear.
