From a technical perspective:
To light up smokeless powder (or any flammable) you need to get enough heat close to a grain of powder for long enough to cause it to ignite. There is a LOT of heat inherent in a spark-gap electrical discharge, but it's very, very localized, and very, very short for a basic static discharge. That "pop" you hear is a mini sonic boom from the air displaced when the discharge flashes a small volume of air to a plasma.
The heat's probably more than enough, but the duration, affected volume and overall energy content of the spark is not enough to raise even a portion of a powder grain to its ignition temperature. If the ambient temperature were very high, it'd be easier, because you'd have the powder grain closer to its flash point. Flash point is the temperature at which the powder will spontaneously combust, with no extra heat input required. By way of example, if you drop a lit match into cold cooking oil (or diesel fuel), it'll go out, because the ability of a small flame to heat the oil on contact isn't enough to raise any portion of the oil to flash point (and release flammable vapors) before the small flame is suffocated by contact with the liquid medium. If you raised the cooking oil to a few degrees below its flash point, then dropping a match in has a good chance of lighting the oil before it goes out. So... doubtful the small spark of a static discharge could raise room-temperature, or even desert-environment-temperature smokeless powder to ignition temp all by itself, even if it could somehow get to the powder through the cartridge case.
Back to how a primer is different - it's a SHOCK sensitive explosive compound. The plasma discharge and supersonic crack of a spark gap discharge occurring are potentially more than enough to set off primer compound, if it gets to the primer compound unobstructed. Once the primer compound explodes, it generates a much larger volume of hot gaseous byproducts, which are then enough to ignite the powder in the cartridge.
Primers would be no good if they were TOO shock sensitive, however, and the primer compound is buffered from the shock of a spark gap discharge from its outside, by the thickness and resilience of the metal in the primer cup. A fat spark discharged into the open end of a cup very well could set off a primer, but I think has about a zero chance of doing so for an installed primer, from the cup end/outside.
There ARE electronically-ignited primers, though they typically use a capacitor discharge, like a camera flash circuit, to boost the duration and temperature of their spark, and insure reliable ignition. If you were to fashion a special spark-sensitive primer for small arms ammunition, it could likely be set off by a fat static spark, or a piezoelectric discharge, like a lighter's, but only if you designed the primer to channel that spark across a gap inside it, and designed the firearm to get that spark into the chamber, to contact it in the first place.
So, in mythbusters fashion, the base idea is BUSTED, because you'll never get a static spark inside a firearm to the primer, or inside the cartridge to the powder, and if you did, unless the primer or something was specially designed to channel that spark inside, it wouldn't do any good.
Onward, in mythbusters fashion, if you wanted to TRY to set off a cartridge with a spark of some sort, then I think you could probably achieve it with a stun gun, contacting the primer, and after a bit, that spark would heat things up enough to maybe set off the primer. Best-case scenario for a try at electrically igniting a handgun round would be to wire a stun gun to the firing pin (insulating it within the channel) so that it'll discharge from its closest point into the primer cup, and then let it rip until the battery runs out or the gun goes off. If that didn't do it, you could replace the contents of the primer with a lower ignition temperature compound, including its own oxidizer, and repeat the experiment to see if the spark can heat the cup enough to ignite its contents, and in turn light off the main powder charge.
FWIW, an electric blasting cap works by explosively detonating a short, thin, length of wire, with a pulse of current, and it's the shockwave (just like that of a static spark discharge) that actually lights off a high explosive, not heat. That's why shooting C4 or dynamite or tannerite will set it off, while setting it on fire by itself will just cause it to burn up slowly.






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