Computers can also include physics based random number generators so that's still simulatable. And analog computing also is possible. Quantum computing not really yet but theoretically possible.

Not sure if any other wizard hat things than those can extend classical digital computing

This is just partially a solution, because this assumes cognition as a whole is computable.

Because randomness does not introduce agency.

Bringing up 'Quantum' effects to somehow free us from determinism and somehow introduce free will, is completely false. Somehow using the word "Quantum" explains human consciousness is wrong.

A coin flip is still determined. You don't know which way it will flip, but also, you aren't deciding which way it will flip. The future is determined either way, determined one way, or determined the other way.

It does not introduce, but maybe the combination of both does (see Searle points in freedom and neurobiology). You also need to prove that the future is determined, simply assuming so is an error.

> you obviously know at the smallest level things happen by random probability

We don't actually know this. It is just what we observe that fits our map so far.

If you don't believe in God, actually you NEED to buy this, or buy into some other methaphysical eternalism that is as hard to prove as truly random events. The universe either started randomly or it was created, and the existence either existed eternally or it was created (both prospects are also problematic in some level, but at least it is something better than thinking you can not have a position on this and also assume that determinism is trivially true).