Given that in shellsort there are many different choices for the gap sequence $($e$.$g$.,$ a series of positive values that decreases to $1$ like $"1,4,13,40,$dots $",$ or $"1,4,9,13,$dots, $,$ according to some formula$),$ how do we know that it always produces a sorted output?

As the algorithm runs, it widens the distances between elements it compares until the entire array is sorted.

The gap sequence means that shellsort will eventually act like insertion sort.

Because the best case performance of $NlogN$ means that the algorithm must produce a sorted array.

It doesn't. This is the reason why the average analysis for its run$-$time is unknown.