That is what Turing complete means. We're assuming that a Turing computer is capable of doing any kind of computation that is physically possible. Of course, it needs a quantum computer to do quantum calculations, so the Church–Turing–Deutsch principle states that it needs a quantum computer in order to truly execute every physical calculation possible, but that's whole another beast. Turing-complete just means minus the quantum processes.

It is possible that the human brain is doing some sort of quantum calculations, but most would probably doubt.

>There is no evidence to suggest a Turing computer can reproduce the "mind", which is really the crux of OP's point.

Of course that there's no "evidence" because we have never created a mind yet. The point is that a Turing complete CPU is physically indistinguishable from the human brain. They are the same thing in principle.

The "magic" is in the programming. We just don't know how to program a mind yet.

The "evidence" is in the human brain. The mind exists inside of the human brain. The human brain is a physical object, just like a CPU is. The human brain is Turing-complete. So is a CPU.