Man-made satellites were put into orbit by the thrust of one or more rocket engines, which continuously changed their orbits gradually all the way up. It was not by a collision or quick change of orbit from a close fly by or a collision. In fact you cannot put an object into orbit with a quick impulse addition at surface level, as the object would return to the same point after one roundtrip around the earth. Unlike man-made satellites, most asteroids do not have that slow thrust over long periods, which slowly change their orbits, unless they thrust something into space, or are close to a massive object at nearly the same speed, following for longer periods. I know that celestrial objects can have an in build “rocket engine” e.g. do comets boil, when they come close to the sun, and Io has geysers that erupt into space due to tidal heating.
Regarding the ejection of planets or moons from stable orbits it normally requires many interactions, so if that mechanism should work in reverse to catch objects, I would expect that it would require many interactions too. Most objects coming at a random speed from deep space I expect would most likely come in at such a speed and angle that they will only get one flyby, and get their orbit changed once e.g. by flying close to a moon and continue back into deep space. For example calculating the gravity assist manouvers to eject something from the solar system is very complicated, and it does not just happen by change that you are close enough to the planets for their gravity field to become significant. Likewise I think getting an object comming in at high speed from deep space and getting it to slow down enough to go into orbit, by it flying close to a planets moons, is very unlikely to happen by change.
If the object got captured by interacting with something else, changing its path from a non-orbit to an orbit, I would expect to see orbits with a shape very similar to non-orbits. E.i. orbits that are far from circular, but instead reaches far into space where the object captured originally came from, and where it got its orbit influenced. I.e. If the orbit was changed by a collision in deep space, the conservation of energy dictates that orbiting object reaches the same point after one orbit. I know that many orbits exist, and non of them are especially magical, but I think it is much easier to achive a non-orbit, than for the speed distance relation to be such that an orbit is achived.