Instead of getting all vertices, you can try the MID option for midpoints of the centerlines.
Then use Generate Near Table tool, which allows you to specify a search distance and a maximum count. So if you specify 2, the tool will find the two closest features within the search distance for each midpoint. Hopefully each midpoint of the centerline will find two distances - one to each sidewalk. The sum of the two distances would be the width at the midpoint of a centerline.