Why do we use dedicated earth pits for lightning protection? Why can't we connect the down conductor to earth pit used for earthing?

Lightning may create huge voltage surges in the system. If dedicated earth pit  used then spantaneous lightning (less than 1.2 micro seconds) will be travelled directly to the ground. Which is also shortest resistance path to lightning stroke/waves.

If dedictaed earth pit not built then it will travel though remaining earth grid. That leads to ground potential rise (GPR) for many appliances/equipment connected to earthing system.

Thus why all down conductors are connected to individual earth pits to create shortest resistance path. These earth pits also connected to remaining earthing system to make redundant path. 

So if in case of lightning, the extra voltage will drop to the applicances, it will drop directly to ground

NEC/NFPA 780 & NEC 70 do not restrict to provide separate Pits for Lightning Protection AND Equipment Grounding. These can be merged in one Grounding Loop, however, we need to provide dedicated grounding Electrodes for each equipment as per its requirements as stated in the NEC 70.

And if you think i am failed in understanding your question, please revert with details.

you can use either dedicated earth pit or the one used for earthing.