That's a very nice experiment.

Not very mysterious, I am tempted to say. At the beginning, seems normal to me for the bulb to light up with a single wire. When we were kids, we used to tune a wire antenna by connecting a miniature incandescent lightbulb in series with the antenna, and adjust the impedance matching for maximum filament heat. The energy flows "through one wire" because it is radiated by the telescopic antenna, radiated as electromagnetic waves.
About the attracted foils at minute 28, my guess is that's the photoelectric effect at work. There might be some ultraviolet light produced, maybe some X rays, too, and those will hit the metal foil, and bump away electrons from the foils, thus making the foil to become positively charged. The rest is electrostatic. Accumulated charges on the suspended foils are making the foil stick to the bulb's glass.
Same thing (photoelectric effect) is what charges the HV capacitor at the end of the video. Note how the spark is bigger when you hold the other end with the hand, providing something equivalent with wiring the other end of the HV capacitor to the earth.
Sure, this can be verified by more experiments, so to eliminate possibly wrong hypotheses. Keep all the variables the same, except the one you want to test for. For example, add extra weight so to make all the foils the same, except the type of metal.
I expect the spectrum from a spark-gap to be very, very broad, much broader than the oscilloscope's (or the probe) bandwidth. Also, 300W is a lot.
Bulb filaments are usually very thin coils (with small capacitance between turns. That might produce huge resonant voltages along the filament. That might pull filament atoms away entirely. Note how the bulb near the foils got darker. Most probably because filament atoms were pulled away and deposited on the interior of the glass.
Another thing is that sparks and/or very high intensity fields make electrons to escape from the wires. Another thing is any acceleration/deceleration of particles produces X rays (Bremsstrahlung Radiation) this includes particles hitting something, turning direction or "jumping away" from a wire (e.g. Corona Discharges) and so on.
In general, electrons don't like spikes, or sharp edges, they tend to fly away from a material with spikes, because other electrons crowds on only one side, and the electrostatic force pushes out the electrons at the tip of a spike (or else said the electric field is much stronger at the sharp points)
I'm no expert in very strong and broadband EM fields, just that it doesn't look like something outside of the known (mainstream) physics.