Well, as I said, it's not so much that we can't grasp what goes on below, so to speak, but rather, that the question just isn't applicable. The models we construct of the world have the characteristic that there exists some fundamental layer, some base facts whence the rest flows, but I think it may be a mistake to conclude that hence, the world itself must have that structure. So while we can answer this question with respect to a given model, the question just is ill-posed if asked of anything but a model.
In a sense, one gets a hint of this in dualities: both gravitational and quantum theories have some 'fundamental' core, but they disagree, in general; nevertheless, in certain cases, both may turn out to describe the same physics. So which of the two theories' fundamental layers is the right one for the physics they describe? In my opinion, since neither is privileged compared to the other, the answer can only be that none is. Each model has a fundament, but the world doesn't. What we see as fundamental is just an echo of the way we build our models.
I think that the appearance of accident and randomness is actually another clue in that direction. One can generally represent a noncomputable function with a computation plus an infinite random string. Consequently, a being building computational models of a noncomputational world would end up describing it in terms of computable, deterministic evolution interspersed with random events---which is of course just what we see in quantum mechanics, and perhaps in the selection of initial conditions and such.
In the end, you may well be right that our approaches could be seen, in some way, dual to one another. It's kind of like asking, 'what exists?', and you answer, 'everything', while I answer 'nothing'. But these aren't ultimately distinct. Likewise, refusing to answer the question of fundamentals and answering it with a certain universality class of models where we can't further find out which is the right one may not be too far away from each other.