If I understand you right, your question is simply this:
since there is external electric field (in your example, the induced field of the torus coil) that does positive work on the charges bound to the secondary circuit (since current in this secondary circuit increases), but the external magnetic field is zero near the secondary circuit, isn't the Poynting vector there zero? How can EM energy move into the wires of the secondary circuit in agreement with the energy interpretation of the Poynting flux from the Poynting theorem?
The answer is that the Poynting theorem does not relate flow of EM energy at some point of space to the external fields at that point (like the fields of the torus coil), but it relates it to the total electric and total magnetic field (sum of all contributions due to all bodies). In the vicinity of the secondary circuit, this total field includes not only contribution due to the torus coil, but also contribution due to the secondary circuit.
As soon as there is non-zero electric current in the secondary (which requires arbitrarily small energy), total magnetic field is not exactly zero in the spatial neighborhood of the secondary circuit wires, because the secondary produces its own magnetic field.
Thus except the time when current in secondary is zero, the Poynting vector near the secondary is not zero. Initially, the secondary current is small, so the Poynting vector is small, so the rate of energy transfer to the secondary is small. But as the secondary current builds up, rate of work of the induced field of the primary on the secondary increases and the Poynting vector flux into the secondary increases as well.
If you're thinking what happens when we suppress the secondary magnetic field too, by making the secondary to be another ideal torus coil with zero magnetic field outside it, then you are right that the transfer of energy will not go well. Of course, real torus coils made from real wire will not have exactly zero magnetic field outside, because there will be stray fields due to concentrations of current in the wires and lack of current in between them. So some transfer of energy will still happen, but the geometry of mutually tangled toruses seems to be inhibiting this transfer. I did not do the calculation, but most likely mutual inductance of such tangled torus coils is very small, because each turn of one circuit has almost zero flux due to the other circuit. The more tightly the coils are wound the lower the mutual inductance and the slower the energy transfer between the two coils.