Addendum:
I am not sure that the process of simplification can help in anything. But I'll try to clarify what I have in mind. Imagine that the bold black curve is the path of minimal length (I didn't draw the ambient polygon). You go from $a$ to $b$ etc. The point $b$ is the first point of "inflection", means that the turning changes sign. If you don't touch the first and last segments on $(a,b)$ you will not change the total turning angle on this piece of path. You have then (on the picture) the choice of three simplifications (thin lines marked with circles at rupture points), two of them gain 1 segment count, and the other one gains 2. If some of them fit in the ambient polygon, it's fine, and you continue that way, on the next part, until the next inflection point. You would like to simplify also by shortcutting along the green dashed line, but you cant since the bold curve is assumed to be the shortest, it must be an obstacle there. So, this way starting with a short curve you may improve the straightness. As I am really not an expert on this kind of geometry, I am not sure this way leads somewhere. It seems that there is still a long way from here to a proof of the existence of what you are looking for.
