Sorry, that doesn't answer my question. The issue IS how a singularity can become a universe that is in fact, non-uniform, without resorting to an explanation that requires an external influence on the distribution of matter from the singularity. Quantum mechanics does NOT explain the resulting non-uniformity. No one has ever adequately explained how non-uniformity can result, but I gave you a shot. I thought I would let you take a stab at it anyway.
Quantum mechanics does NOT explain the resulting non-uniformity. Yes, It Does!
Let's assume a totally uniform distribution of energy. With a high enough energy density, particles will be pulled out of the vacuum. This is an intrinsically random process; the particles will not appear simultaneously on the points of a lattice*, but will have a random distribution. Random distributions by nature have areas of differing density. As in the distribution of cancer cases in the U.S., the dense lumps don't require a cause. Any resulting lumps will be amplified by the effect of gravitational collapse.
*Moreover, even if the particles did appear on the points of a lattice, there may still be a "handle" upon which gravity could act, because you cannot have a lattice that is free of multipole moments to all orders. The resulting non-uniformities may be of regular size, shape and spacing, but they'd exist. It all comes back to the fact that a particle, by definition, is itself a localized (read: non-uniform) concentration of energy.