I don't think you can apply this proof to the described algorithm.

This proves that

    for i in range(min_key,max_key):
        swap(arr, i, choice(i, max_key))

works as a shuffle, by growing a random sub-array.*

The first algorithm was

    for i in range(min_key,max_key):
        swap(arr, i, choice(min_key, max_key))

which has different biases.

* I am not actually sure that this actually works.