Its gravitational field is so huge that it is basically the vacuum cleaner of the solar system. Also a lot of things that are not in orbit of it are occasionally significantly perturbed in their solar orbits, by Jupiter, such as a number of long period comets.
An orbit can't just change from a parabola into an ellipse, without the influence of external forces. Is something slowing down these object so that Jupiter "catches" them? (Rings?)
You're right that in a two body system this is totally impossible. But for a three body system of Jupiter, the Sun, and our new moon then ballistic capture will baaaasically get you into orbit and then it only takes a small further interaction with another planet, another moon, or whatever to get into orbit the rest of the way.
The important thing, as far as I understand it, is the way the future moon slowly transitions from being primarily under the gravitational influence of the Sun to primarily under the orbital influence of Jupiter.
We've actually used ballistic transport to transition probes from orbit around the Earth to basically in orbit around the Moon with just a tiny braking burn at the far side instead of the serious burn that a Holman Transfer orbit would require. And the Moon is in a stable orbit around the Earth and doesn't have any further satellites.
There is no slowly for capturing an incoming object - you get one chance, unless that object is in a similar orbit to your (and in that case the Sun doesn't help).
> with just a tiny braking burn
Tiny is relative here. It's still enough of a burn to significantly change the momentum of the rocket. Moons don't have anything with that kind of energy, except other moons.
No, slow is a fair description of this. You need to start out in nearly the same orbit as the target and while you're both traveling around the Sun or Earth fairly quickly you approach the target much more slowly than you would in a Holman Transfer. Then when you're captured you'll be going around the target a number of times before other gravitational perturbances eject you again, giving a long time to do a burn to get into a stable orbit. So yes you have to change the momentum of the rocket but the burn could take place over the course of a week and won't have to be very large. Your burns to achieve an intersect giving you a ballistic capture would typically be much larger than with a Holman transfer but the slowness of the final approach means you're free to use an ion drive, solar sail, or other high efficiency and low thrust drive.
Its gravitational field is so huge that it is basically the vacuum cleaner of the solar system. Also a lot of things that are not in orbit of it are occasionally significantly perturbed in their solar orbits, by Jupiter, such as a number of long period comets.