It's a state mandate, not by district.

Only 3 states have a cut off date that goes past Sept 30th and one of them is Oct 1st.*

This date was not randomly picked by some random person, it is research based and enforced for a reason.

*There are 3 or 4 states who let local districts choose the cut off dates, but it looks like these are all states that do not require Kindergarten. (New Hampshire also does not require Kindergarten.)