But what is the actual physical mechanism (for the lack of a better word) why this extra current must be drawn by the primary?
The energy put into the coil can creates a magnetic field. If this coil were alone that's all it would do. Because there is another coil, the magnetic field does work (transfer of energy) on the electrons in the secondary and can create a voltage (to do more work in another part of the circuit). However, if there is no load on the secondary then no work is being done and the electrons don't move. Voltage is still created but it's as if the secondary isn't there and the transformer behaves more like an inductor, only generating a magnetic field.
So what is the physical reason why more current "has to" flow into the primary when current is drawn from the secondary?
Because the electrons in the secondary are under the influence of a magnetic field, they want to move. This will create a voltage (and current). The models neglect the magnetic field (usually) because most time series simulations only use voltage and current, calculating the magnetic field is unnecessary for most people.
Coils, turns and other variables are just ways to simplify what is really going on. Each moving charge carrier (electron) creates a magnetic field and that magnetic field can move other electrons. Since we don't want to calculate the field for every charge, we use vector calculus to simplify this process with these laws: Biot Savart and Maxwell's Equations and Faradays law of induction. These are then simplified to come up with the transformer equations