There is minimum amount of energy from falling down Earth’s gravity well. Earth escape velocity is 11 km/s. A big chunk of asteroid energy comes from Earth’s gravity.
On the other hand, the orbital speed of Low Earth Orbit is 8km/s. Thus, with regard to asteroid redirection missions, we'd need 'only' about 3-4km/s of Δv to prevent this 'gently setting down' asteroid from hitting the Earth by capturing it into orbit.
Definitely into science fiction territory here considering that DART resulted in a Δv measured in centimetres per second, but I'm still rather tickled by the idea of collecting a new moon for ourselves :)
If we don't need to capture it to orbit, but just avoid hitting the Earth, centimeters might be enough. The trick is to be able to do it way in advance.
Besides, why LEO? Even GSO is 11 times closer than the Moon.
You're quite right - attaining GSO would indeed require only another 3 or 4km/s. But it does seem a little unfair to put it in GSO; we'd surely want people in both hemispheres to get a chance to gawk at our new satellite!
