a simulation that selects an Everett branch at random from a distribution given by the Schroedinger Equations

At which moment do you select the random branch?

The equations say that each branch has an "amplitude"; and unlike probabilities, these amplitudes are complex numbers. Which means that two nonzero amplitudes added together can produce a zero outcome. (As in: "yes, it is possible that 'A and X' happens, and it is also possible that 'A and not X' happens, but 'A' is completely impossible".)

This effect almost completely disappears at larger scale, because what we define as a "state" on the large level includes zillions of possible states on the small level (for example, the state "the chair is in the middle of the room" corresponds to zillions of possible combinations of particles), and these large states interact just like classical probabilities, because of some mathematical laws about large numbers and complex numbers.

So yes, on larger scale we can pretend that the world is randomly selected from all the possible branches. It just doesn't make sense on the small scale... which is exactly how the quantum effects were discovered, because we were originally thinking in terms of probabilities, and then we couldn't explain the double-slit experiment.