[tsimionescu]

But that's the opposite direction from Turing's intention. The point of the Turing machine model is for the machine to be both mechanically plausible (no magic required) but also equivalent to what a human does when performing computation.

The Turing machine model is a mechanically plausible abstraction of a human performing a computation by following some steps in their mind and with a notebook. The tape stands in for the notebook. The read head stands in for the human's eyes. The write head stands in for their pencil hand. The internal state of the machine stands in for their mental state. At every step, you either read a symbol from the notebook tape and change your mental state in relation to this symbol, or write a symbol based on the current mental state. The procedure itself can be written on the tape, and you can occasionally refer back to it.

The original paper spends a good few pages working out this metaphor and showing that the machine model perfectly abstracts the mathematician's work of computation.