Why do we should not to use simple count instead of cumulative count in Counting Sort?
It is related to the stability of output this algorithm:

Stable sort: For same number appearing more than once in the input array, ties are broken by the rule that whichever number appears first in the input array appears first in the output array.

•The property of stability is important only when satellite data are carried around with the element being sorted.

•Counting sort is often used as a subroutine in radix sort. Counting sort’s stability is crucial to radix sort’s correctness.

Having a cumulative sum helps it in making it a stable sort.