How can the local feature size become arbitrarily small?
As you have suspected, it is impossible for a simple smooth closed curve to contain the line segment from (−1,0) to (1,0) such that its local feature size at $(0,0)$ is arbitrarily small.
On other hand, it is easy to draw a self-intersecting smooth curve with that property. Here is an example.


In fact, for every smooth curve that contains the line segment from (−1,0) to (1,0) and intersects itself at $(0,0)$, its local feature size at $(0,0)$ will be arbitrarily small.