In the steady state, DC, does energy flow in the wires or in the fields?
Start with the fact that energy is contained in electromagnetic waves. If you have used solar panels, you should believe this is true.
Consider a plane wave propagating through free space. We know that energy is flowing with the plane wave. We can measure it, and we can calculate it. And the energy flux agrees with the Poynting vector. We don't even need the Poynting vector to calculate it, but it does agree with the calculation:
https://phys.libretexts.org/Bookshelves/University_Physics/Book%3A_University_Physics_(OpenStax)/Book%3A_University_Physics_II_-_Thermodynamics_Electricity_and_Magnetism_(OpenStax)/16%3A_Electromagnetic_Waves/16.04%3A_Energy_Carried_by_Electromagnetic_WavesNow consider a transmission line. It can be a twin lead transmission line as we have discussed. Let's just feed it with a lumped source with say a sine wave. Again we know energy is propagating. We can measure it and we can calculate it. We know the wave fronts are moving and we know that since there is energy in the field, the field is carrying the energy. We can't say the energy flows in the wires or we would count twice the power that we measure.
Now lets say instead of a sine wave, we just have a battery and a switch. At time t=0 we turn on the switch. We have a rising edge, say 0V to 5V. Now we know that this edge propagates down the line. We can measure it. We know that the energy is moving down the line in the fields. We can see that if we slice up the space into thin slices of width dz, then as the wave front arrives at a point on the line, the energy of the slice goes up, and we know that that energy has to come from the slice behind it. So one by one slices are filling with energy and energy is flowing all along the line in the fields.
Now if we look at a point where the wavefront has already passed, the fields are constant. The voltage is constant and the current is constant DC. But we know that energy is flowing through this point to fill the slices up to the wave front. So here everything is static, except power is flowing through the fields.
Now we terminate the line with a resistor that matches the impedance of the line. When the wavefront reaches the resistor it is totally and continually absorbed in the resistor. But nothing has changed at the previous point on the line we looked at. Energy is still flowing there in the fields even though the whole system is now in a steady state and DC only.
That's one way of looking at it.