To make smoke, oil just requires heat. So if the primer did a slow, cigarette-type of burn and there was still oil in the perimeter crack where the primer meets the sides primer pocket, that could smoke. But more commonly in squibs, what has happened is pressure got just high enough to expand the case just enough to let some smoke out around the case mouth that comes back around the cartridge and out of the chamber. Of course, it's also in the muzzle, but that's a longer path so that may not show up immediately. Pierced primers also can leak smoke at the edges without the pierced location being immediately apparent (usually covered by soot).
The Box-O-truth test is a little funny. The second test on live ammo uses ammo that has sealant applied to the primers. In the end, the test may only prove the primer sealant is effective. Do that with ammo without primer sealant or, better yet, a loaded cartridge case that has been reloaded a dozen times and has scuff marks up and down the sides of the pocket and no primer sealant, and then you may get a different result.
In the first B.O.T. test, he soaks unnamed primers and succeeds in killing them. But primers are different. If he'd done that to Federal primers that have a big blob of sealant dried on the inside of them, would he have had the same result?
We've had posts of primer killing tests before, and the results have always been mixed. Some are, and some are not killed by soaking in various things, including WD-40, IIRC. If it interests you, try searching the forum for past posts on the subject.