[formerly_proven]

You can explain this experiment just by utilizing basic transmission line theory (which is not a simplification per se: transmission line theory is just the observation that you can easily derive the behavior of E and M fields for some conductor geometries, like in Veritasium's video).

The long wires act as transmission lines with an easily calculated impedance of ~1 kΩ. When you close the switch, you generate a pulse - a very broadband AC signal - which starts to travel around the circuit. While the pulse is traveling in the line, the line's inputs see the line impedance. So you get the car battery in series with 2x the line impedance in series with the impedance of the light bulb. This directly tells you the amplitude you're going to see near-instantly at the light bulb. Light bulbs are usually low-impedance (e.g. a 10 W, 12 V light has a hot resistance of around 14 Ω), so you're only going to see a tiny pulse on the light bulb. After the pulse has traveled the length of the line and back the impedance seen across the line inputs is the short at the end in series with the conductors resistance of the line itself; in a heavily mismatched setup you're going to see some more reflections going back and forth, this causes the stair-stepped rise over multiple bounces when you do this experiment with "realistic" values.

It's basically a slightly unconventional TDR - time-domain reflectometry - setup.