I'll repeat the comment I made on the video (which I'm sure has been utterly buried under a torrent of less extensive comments, or suppressed outright by algorithm):

The way I see it is structural:

He's using a DC circuit in an AC field, and claiming that the DC circuit still holds, while forcing an AC behavior upon it.

As is always the case in proof by contradiction: we have only proven that our premises were wrong.

It's a similar fiction as the conservation of charge vs. energy when connecting charged capacitors together: if energy is conserved, where does it go? Well, it turns out that you can't simply short capacitors together without taking account of their resistance, or inductance. Or, more generally, of the loop area between them, which gives rise to both elements. So the charge is conserved (a more fundamental quantity), and energy is conserved whether or not you've written in a way for it to do so.

It would be more illustrative if he phrased it as a riddle to the student, to figure out where the disconnect is.

You are quite correct that merely adding a transformer, and keeping track of the probe wires in the field, is all that is missing!

As for failings of Kirchoff's laws: radio waves. One must get ever more particular about where (spatially speaking) one applies them. The current flowing into the feedpoint of a dipole antenna, for example, does not equate with the current conducted out of the element tips, which is zero. The disconnect here is concentrating on conduction, while ignoring displacement current. In effect, equivalent capacitance carries the current into free space. But more accurately, it's carried into the fields around every point of the antenna.

At their most general (but perhaps least useful), KVL/KCL reduce to a single point only: they are a differential relation, which must be integrated over the space of interest. (This, of course, is painful to do by hand for all but the simplest arrangements, so we usually have computers divide the space into millions of finite elements and apply the laws to them, for us. Hence, FEA (finite element analysis) tools.)

This, in turn, drive home another point about schematics: what we draw is an abstraction, a fiction, a model. The points are connected instantaneously in time and space, with no concept of distance, or the speed of light (namely, that the distance is effectively zero, or the speed of light is infinite). To build a realistic model, when the speed of light is relevant, we must introduce enough parameters (whether L and C approximations, or real transmission line elements) to match this.

Tim