Electro-magnetism (E&M) is genuinely weird. Most people never find this out, because most people never go into electrical engineering or a physics PhD (where you really have to grapple with it).
Most people think of electricity as volts, amps and watts. Maybe ohms. We don't use ohms in a household context.
The initiated talk about capacitance, reactance, inductance, resistance and conductance. They talk about "transmission lines", "skin effects", and "antennas". The rest of us need to be electrical engineers before all that makes sense. (Oh. Wait. I almost was one.)
Here's a way in: metals like copper are often called good "conductors of electricity", as if electricity is something that passes through the metal. Instead, think of metals as good receivers of electromagnetic radiation. Wires do not in some sense channel or concentrate the electromagnetic fields, or act as pipes for electrons to flow along, they respond to the electromagnetic fields. Indeed, everything responds to electromagnetic fields. Mostly not much.
Wires respond by creating their own little electromagnetic field around them. Most materials (because "everything is a capacitor") respond by retaining tiny, tiny amounts of charge which they then eventually let go of. Air does this. So does polyester, which is why it crackles when you take it off.
Mr Veritasium set up a circuit with a battery, a switch, a light opposite the switch and some very long wires connecting everything. The idea was that the wires would be so long it would take a noticeable amount of time for the electricity to "flow" along the wires and power the light.
Except the light comes on instantly.
His explanation uses a thing called the Poynting vector. Do not use those words near physicists, as they may call your bluff.
Electromagnetic waves have an electric field (E) and a magnetic field (B) that are always in phase and at right angles to each other, and to the direction of travel of the wave. (This is why there have to be at least three physical dimensions, or we couldn't have Radio Three.) Since electromagnetic waves carry energy, it makes sense that there should be an energy vector corresponding to the wave in the direction of travel. Poynting proved that this vector (*) S = E x B, where 'x' is the vector cross-product, and I've left out a constant of proportionality. It's the E times B bit that is the achievement, not the direction (cross product), because we got that already from the physics.
So Mr Veritasium said, the electric field is pointing this way (points along wire) and the magnetic field is pointing that way (points upwards) so the S vector must be (pointing at the light bulb). Presto! The energy gets to the light more or less instantly.
Which convinced absolutely nobody, because they piled in to discuss this using anything but Poynting vectors. Transmission lines and displacement currents was a favourite, because, well, engineers. Nobody was doubting the Poynting explanation, because physics > engineering, but because they were engineers, they wanted to explain it in terms of something more familiar and material.
Complete the well-known phrase or saying: cart, horse.
Poynting's insight was that the materials in which the wave moves (e.g. air, wires) do not facilitate the power transmission, rather they modify the electro-magnetic waves, and hence the power transmission. The transmission-line / displacement current explanations are consequences of the transmission of power in the direction of the Poynting vector, not explanations. When modelling a specific setup, the B vector (which is for free space) is replaced by the H vector, which takes into account the effect on the B vector of the materials involved.
What happens is this: when the switch is closed, a voltage pulse starts to travel round the circuit. This creates a magnetic field B through the Maxwell equation (with J = 0) for B
$\nabla \times B = \mu_0 \epsilon_0 \frac{\partial E}{\partial t}$
which creates a Poynting vector S. Behind that pulse comes the first lap of a current J that will be circulating once the circuit is in a steady state. That sustains the B field by the Maxwell equation (with $ \frac{\partial E}{\partial t} = 0$) for B
$\nabla \times B = \mu_0 J$
which sustains the Poynting vector S. (The E field is sustained by the battery voltage). That S field carries the energy that excites the molecules in the wire in the bulb and creates the light. Because the wire in the bulb is a good receiver of electromagnetic radiation.
Not a transmission line in sight.
It's worth noting that if the bulb was put, say 300,000,000 metres away from the switch on the opposite side of a loop, then it would take 1 second for the bulb to light, but that would still be faster than the roughly 1.6 seconds it would take for the voltage / current wave-front to reach it.
This is, of course, handwaving. More precise calculations would take account of the dielectric air between the wires to calculate H and also factor in the displacement currents, but the principle remains the same. That would start to sound like engineering. But the engineering is there to help perform the calculation, not to help understand what's happening.
(*) Strictly E, B and S are not vectors, which are 1-forms, but flux densities, which are 2-forms. This is the only time in your life you will ever read that.
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