The hypothesis that the nail either punctures on the first rotation, or not at all, misses that something is different on subsequent rotations, namely that the

The hypothesis that the nail either punctures on the first rotation, or not at all, misses that something is different on subsequent rotations, namely that the nail experiences centripetal acceleration once it is embedded in the tyre. You could imagine that on first contact, only the tip of the nail sticks into a tread block. Then as the wheel rotates, centripetal acceleration causes the nail to be pulled more perpendicular, and on the second rotation it punctures the tyre when it hits the ground. No idea if this is actually a common failure mode, but it seems plausible at least.