I can't get my head around the washer and rod exercise without leading to a circular argument, at which point my brain shuts down and spits out the most recent result with an error flag set. The results are the same either way. I cannot see any point in considering the washer and rod simultaneously if I know they are independent. The power that's stored in any magnetic field may come out of it and go back in somehow, but it has no known effect. (Plus we have the benefit these days of knowing what makes the magnetic field, and being able to test the 'moral' reasonableness of any weird situations.)
So roll with that idea, and don't just consider the washer and rod separately, but split them into infinitesimally small threads, or in the case of that example, sectors. This alters the electric field, but has no effect on the potential difference that drives energy transfer (work function in a conservative field). Compute the Poynting vector of that and superpose. Which should be possible if the energies add up. The Poynting vector should then show something quite different.
Now apply that to Veritasium's example (at DC), which is something I did think of earlier but thought it mightn't work (as in, be a reasonable partitioning of the current flow).