I may have been assuming that it was clear in the article, but to help show the effect, the diagrams show only the sound emanating from the plane at one instant in time. In reality, the aircraft is continuously moving, and there's a continuously changing "where is the sound coming from" vector. The main idea is that this "where is the sound coming from vector", if you will, may be behind the plane's light vector (which we can just say is the plane's position) if you're some distance away, leading to this oddity.
> the sound may be able to reach you much much earlier than the plane
I completely agree! I didn't mean to say that a plane's sound is always behind it — it very much depends on the position of the observer. If a plane is flying in any other direction than perfectly perpendicular, the math and the effect will be different.
For "If the plane was moving very slowly, it wouldn’t outpace its sound by much.", I meant in the sense that we, as observers, are perceiving the sound. More that the plane's position vector wouldn't outpace the "where is the sound coming from vector" by as much (a smaller X in the diagram, if you will), leading to "where is the plane" being closer to "where is the plane's noise". Going faster than the speed of sound leads to all sorts of very interesting questions, but I don't believe it would affect this in the simple case we're looking at.