Enlarged text mine.
You've found comfort in a text from 1941? Ok, whatever makes you sleep well at night!
Dude, that's Stratton. One of the most cited textbooks in EM.
Do you think Poynting theorem has changed since then? No, it hasn't. In the context of classical electrodynamics Stratton, Schwartz, Panofsky & Phillips, and Jackson are "the" references. I've heard good things about Zangwill, "Modern Electrodynamics" - I have not read it thoroughly, only a few pages here and there, though - but even if it is a 2012 text, do not expect it to say anything different from Panofsky Phillips or Stratton.
For example, Zangwill says that "
The analogy between this equation and the continuity equation [...] reinforces the interpretation of the Poynting vector S as a current density of electromagnetic energy."
Regarding the circuit with a simple resistor it surprisingly goes even further: "
It is clear from the diagram that energy flows out of the point dipole source, into the vacuum, and into the wire at every point along its length."
But he acknowledges the same points made by Stratton in 1941 and has a little paragraph that says "
It may seem odd that the Poynting vector for a wire circuit does not predict energy flow parallel to the wire itself. This and other unanticipated features of some Poynting flows prompt some authors to define a Poynting vector using S = E x B/mu0 + curl X. The vector field X is chosen to make [the Poynting vector] point in more “natural” directions. The [above] definition does not disrupt Poynting’s theorem because the latter contains only ∇ · S. Relativistic considerations constrain, but do not completely eliminate, this arbitrariness in the definition of S.
There is no real problem in any event because the Poynting vector is not an observable."
Not different from what Stratton wrote in 1941.
And Zangwill is well aware of the progress made in physics since Pearl Harbour.
Point is: you have to choose a theory and stick to it.
Do you use voltages and currents, and consider current as a flow of charge carriers? Stick to classical ED (it automatically incorporates special relativity) and learn how to correctly apply Maxwell's equations (to the point of being able to tell that the Poynting vector does not point directly towards the battery - or even good conducting wires - at DC because you've seen a picture on Feynman where it points radially in, in a resistive wire).
Do you like to think of electrons in circuits as waves of probability? Stick to non-relativistic quantum mechanics. But forget about describing electrons moving inside the conductors and banging ions: it's a wave that's being scattered by the lattice potential. Then that parts in your videos where you say "engineers knows very well that..." becomes all false. Try to use QM consistently to describe EM phenomena and see how far you go in even the simplest circuit.
Are you ready to forgo semiclassical theories and wanna go full quantum? Then stick to QFT and start computing probabilities for everything. But then basically everything you said in your video becomes lies.
You can't have cake and eat it.