The currently accepted answer is entirely wrong. It says
Black holes don't start from a point in (for example) the centre of a collapsing star and grow outwards. It's actually the opposite - the event horizon forms outside the collapsing star.
The idea seems to be that the matter that initially collapses to make the black hole doesn't pass through the event horizon and so doesn't "freeze" there.
None of that is true. In an isotropic collapse, the event horizon does start from a point in the center and grow outwards (at the speed of light). In a general collapse, it starts at a one-dimensional set of spacelike separated points called the crease set. In any case, all of the collapsing matter crosses the horizon and in principle appears "frozen" near the horizon forever. (In practice it can't be seen because the redshift is far too large.)
The event horizon is the boundary of the black hole. Like any boundary, it has no gaps in it, because a gap would make the division into inside and outside meaningless. Everything that ends up inside the black hole crosses the boundary.
black hole entropy conditions and no-hair theorems are asymptotic in nature [...] Since then I've been wondering whether singularities are ever really created, and if not, why do we worry about naked singularities?
Technically, everything is asymptotic in nature. If you make waves on the surface of a perfectly still lake, the amplitude of the waves dies down over time but the lake will never be precisely still again. The settling-down time of black holes should be viewed in the same way. This has no bearing on whether the bottom of the lake exists, nor on whether the interior of a black hole exists.
Also, naked singularities are by definition singularities that aren't hidden by an event horizon, so the behavior of the horizon is even less relevant. My impression is that "naked" in the question is just an intensifier (especially given the title), so I'll ignore it.
to an outside observer, it takes an infinite amount of time for the singularity to form. In other words, it never happens.
For a classical black hole (with an event horizon), you can pick a time coordinate that respects causality (anything that happens at $t_1$ can only affect what happens at $t_2$ if $t_1<t_2$), and that covers everything that happens outside of the hole into the indefinite future, and that doesn't cover the black hole interior, or doesn't cover a part of the interior that includes the singularity. From the perspective of that definition of time, there is never a singularity. In effect, the spacetime never gets around to collapsing that last little bit.
But all that you've really done, if you do that, is fail to cover a portion of the spacetime with your coordinate system. That doesn't make it go away. Even if you aren't interested in what happens there, other people are, and the problem isn't solved for them.