Say we have pointed conductors connected to the top of the high rise buildings. Will the strikes hit the nearby ground in such a case?
A lightning conductor on top of a high-rise structure is designed to minimize the possibility of lightning hitting the structure by the virtue of its pointed tip giving rise to very high electric fields (the reason). However this doesn't always ensure that lightning will definitely strike the conductor. This link has quite a few examples of the same.
This is because when cloud-to-ground lightning happens, the lightning goes the way of strong electric fields. Even though the pointed tips of the lightning conductors set up huge electric fields, the path in which the lightning conductor is in the way isn't necessarily the most preferred path for the strike to happen. So the nearby places can get hit too, as demonstrated in these images.
It might, if the path of the lightning does not get close enough to the rod or its grounding structure in comparison to alternative targets on the ground.
The lightning strokes originate at the clouds, because the clouds have much higher charge concentration and much stronger local electric field. In comparison, the density of charges, induced by the clouds on the surface of the earth, is relatively low and the field is relatively weak, even around sharp objects like a lighting rod.
So, under typical conditions (no mountains or skyscrapers), the origin and the path of a lightning leader is primarily defined by the location of the cloud and not the location of buildings with lightning rods.
If the distance between a descending leader and a nearby grounded structure is similar to the remaining distance to the earth, chances are the lightning will hit the structure. Otherwise, it is likely to hit the earth.
Although the sharp tip of a lightning rod provides the easiest path to the ground, its action is mostly local and is unlikely to change the course of a leader at distances comparable with its height.