Asteroid 2009 BD passed between Earth and the Moon at a distance of 346,000 km. Scientists concluded that if it were to impact Earth's atmosphere, it would cause a multi-megaton explosion in the atmosphere killing everyone within a 10 - 20 km radius, and this rock was only about 10 m across.
I understand that this data is for an asteroid impact, and not an aerocapture, but for an asteroid to be caught in Earth's gravitational field, it would need to dig very deep into our atmosphere, so needless to say it would be a spectacular event, one I would not want to be directly underneath. It would involve much fire (something I like) and falling pieces of asteroid (something I don't like) as pieces would break off during atmospheric entry. Also, asteroids have the unfortunate habit of exploding in the upper atmosphere, and even laypeople know what that looks like.
(Just the first paragraph)
The speculation from the original question seemed to point to the fact that the asteroid would need a few passes through our atmosphere to slow down enough, so chances are we would witness this several times (unless it exploded). Also, if this theoretical asteroid was big enough to be interesting, the results could also include ablation (Dissipation) of a good portion of our atmosphere. NASA is so afraid of this that scientists have devised some interesting methods to stop an asteroid from colliding with our atmosphere, something most people can agree is a good use of tax dollars :)