I think that's missing an important thing. No doubt we are all mostly agreed that there is some fields stuff going on before the wires are connected, but what it's really about is after that, when there is a solid wired connection. Does the energy flow in the wire, on the wire (skin) or is the wire merely a guide and the energy actually flows still in the field? As I see it, and it's sometimes tricky to remember what the argument is about, it's that last option which is the crux of the video and this discussion.
For me the question is entirely about DC and energy inside vs outside the wire. Nothing to do with switches, transmission lines, capacitors, inductors, transformer theory, antenna theory etc etc.
For steady currents you can calculate energy density (ignoring prefactors and constants of nature) as qV + I*Phi [Phi == magnetic flux]. or E^2+B^2 and you will get the same answer. The former describes the electric energy in terms of charges, the latter in terms of fields. You can't really get away from describing the magnetic component in terms of some field in free space because there is no scalar magnetic potential, so I have picked a form where the current plays a role, but I don't need to refer to equivalent circuit elements like L.
Taking only the electric component qV, that is zero on the interior of a conductor because the net charge density is zero. There is a small electric field inside the wire to overcome the wire resistance but the net charge density is zero. The only place with a net charge density is the surface of the wire, and in the charge model that is where all the electrostatic energy is stored. Even though the current is uniformly distributed across the wire cross section there is no (electrostatic) energy density there.
The magnetic component is harder to nail down. For the field centric approach it's no problem: B is unambiguously defined everywhere, so we can just integrate up B^2. But the flux * Phi representation is sort of inherently non-local: its is the current around a loop times the magnetic flux through the loop, so a product of quantities measured at two different locations.
So at DC, you can consistently define the electric component of the energy density to the wire *surface* as an alternative to the fields. When you include the magnetic component or deal with AC or transient behavior you pretty much have to fall back to a field based approach to energy density. There is no reasonable way to quantitatively define the energy to be stored in the volume of the wires.