Good article, but there are few miss understanding there.

First, you are mixing things, binary_search is just binary_search, you are adding a sort function inside of it. You are mixing responsibilities.

Binary search expects a sorted array, it’s not responsible to sorte the array.

About the sort: The quick sort algorithm is O(n log n). if python is using it by default which is already greater then O(n).

Second, what you describe is the behavior of a O(logN) and O(N) if you put it in the graph you will see. If your N is small you might have better perfomance with O(N).

Third Thing: BigO is to check the behavior when exposed to large scale. In small scenarios, the inner CPU instructions also counts (comparisons, if, etc)