Thanks! - this discussion has led me in directions I have not given any thought to before, and maybe it's not finished...

The questions of what it means to understand something and what constitutes a good explanation seem related, and are both issues that I wish philosophy of the mind would spend more time on (but I'm probably just reading the wrong authors...) Formally speaking, science is not concerned with either of these issues, yet, paradoxically, it seems to have vastly expanded humanity's knowledge of, and ability to explain, the natural world.

This is off the top of my head, and perhaps it is an idea that's already been thoroughly trashed, but suppose understanding is a matter of developing intuitions that work? Until we have those intuitions, we have no idea what's likely to happen next, and that makes us feel uneasy and unprepared, but once we do, we feel more certain that we can handle whatever happens.

Putting aside our intuitions about other people, which are a huge part of being human, most of our other intuitions about the natural world are in Freeman Dyson's second layer - the layer of tangible, rather than fundamental, things. If we spend some time boating, for example, we develop intuitions about what might go wrong in stepping from a dock into a small boat. One way - perhaps the only way - to develop intuitions about things on the first layer is to work with them enough that we can begin to anticipate how things will work out before we have worked through the analysis (I say analysis, because, as Dyson is pointing out here, we have reached the point where most of our fundamental knowledge is abstract.)

This is not the whole picture, however, as there are many things, such as the Poynting vector, that are neither fundamental nor tangible. I imagine that if you work with the Poynting vector enough, you will develop intuitions about where it points before you have done the math. If you can look at a situation and say something like "Well, the Poynting vector will initially go this way, so there will be current in the load as soon as L/c seconds of the switch turning on", then you have provided an explanation (not the explanation) that works for yourself, and also other people with a similar familiarity/understanding of the Poynting vector.

At this point, you may well be thinking that this shows Veritasium was wrong to talk about the Poynting vector, as, presumably, few in his audience have this sort of familiarity with it (I don't.) Personally, however, I feel that what he was doing was useful, and also a very common, and perhaps unavoidable, pedagogical approach: he seems to be saying, in effect "it so happens that there is this abstract concept, that stands in this relationship to electromagnetic fields [also abstract concepts, by the way, but ones with which his audience are presumably more familiar] and which answers the questions we are concerned with here: how does energy get to the lamp, and when does it get there?" It is not a from-the-fundamentals, layer-one explanation, but, unless we are being unduly (and probably inconsistently) skeptical of science and scientists, we can be confident that it can, in turn, be explained from the fundamentals. Transmission-line theory is even further removed from layer one, yet it similarly offers a stepping-stone to it. These are useful abstractions that save us from figuring out everything from first principles all the time, which would be impossible for most people (such as myself), and inordinately burdensome for those who could.

I am not sure that Maxwell's theory works in the mechanical domain: while there are variants of the wave equation for transverse waves in solids and along material boundaries, is there a material or a mechanical analogy to the perpendicular interacting electrical and magnetic fields of Maxwell's theory? (Maybe there's a hybrid situation with piezoelectric materials or materials that do not follow Hookes law?)

I take your point that there are are analogs of the Poynting vector in other domains, and I assume you are raising this in support of another claim that I am now willing to provisionally accept: that there could be a valid and useful mechanical analog for Veritasium's experiment - it's just that I haven't seen one, yet, that I feel works well. The thing that makes the Poynting vector unique among its analogs is that it is the one in the specific domain that we are concerned with here. The other analogs might be just as abstract and difficult to develop intuitions about, or, if not, to relate to EM.

If my story in my prior post about Maxwell's notebooks is correct, it could be seen as suggesting that mechanical models help, as they certainly seem to have helped Maxwell - but if they are still useful now that we have the theory, I would expect them to be used to help teach it now. There's a cautionary tale, on page two of Dysons paper, about analogical reasoning, in his account of Maxwell's address to the 1870 annual meeting of the BAAS. As Dyson tells it, most of that talk was ostensibly about Kelvin's analogy between molecules and vortices - an analogy that did not lead to any physical knowledge.