Consider the randomized algorithm A: compute the state of the universe at time t, then sample a classical configuration with probability proportional to its squared inner product with the universal wavefunction.

Consider the randomized algorithm B: compute the state of the universe at time t, then sample a classical configuration with probability proportional to its inner product with the universal wavefunction.

Algorithm A is arguably far, far simpler than Algorithm B, because the component 

probability proportional to its squared inner product with the universal wavefunction.

is arguably simpler than the component

probability proportional to its inner product with the universal wavefunction.

The difference is the simplicity of normalization, which you need to perform in order to find the probability density. If I recall correctly (and see reference below), normalization of the classical wavefunction satisfying the Schroedinger equation is relatively easy with respect to squared inner product (modulus squared), because all you have to do is find a single constant which normalizes the wavefunction at any particular time (your choice). Once that has been done, then the wavefunction remains normalized forever, with respect to the modulus squared, i.e., with respect to Algorithm A.

I haven't checked the math, but I would be flabbergasted if normalization with respect to Algorithm B were anything like that simple. On the contrary, I would expect to need to find a new constant for each moment in time.

As long as we are reasoning from simplicity, which you seem to be doing, then this seems to provide us with a strong reason to favor Algorithm A over Algorithm B.

reference:

if a wave-function is initially normalized then it stays normalized as it evolves in time according to Schrödinger's equation.