Using random numbers affects the condition about determinism not computability. If universe is not deterministic then the choice made does not depend only on the human making choice but also on the state of generator that creates random numbers. Of course even here it is possible that the state of human is organized in such a way to change probability distribution to not depend on randomness, but that becomes equivalent to deterministic universe with additional complications.

The computability is relevant to the argument because it allows us to create simulation, and to observe a universe from outside. If it was not computable, say required real numbers with infinite precision, and the finite approximations were not able to describe complex things such as humans, then we would not be able to complete our thought experiment.

> The very action of pausing or modifying the universe from without would make the universe non-deterministic.

There are two parts, the starting state and the evolution rule. Here the evolution rule is still deterministic, and the requirement is for it to be able to continue from different starting states.

> Who's choice? If the algorithm is making the choice then the humans simulated by such an algorithm would not have any more free will than a video game NPC.

If human is a part of universe, and does not have a soul, then he is equivalent to its starting state plus the algorithm. If the computation cannot be reduced to a simpler algorithm then no matter how you compute the future state you get that human thinking, feeling and making a choice. The difference with game NPC is that the algorithm doesn't have a hardcoded set of inputs and outcomes but can accept any inputs and produce outcomes that can't be predicted by anything other than that algorithm with that starting state.