If a script is listed under the "Remove Unused Javascript" tab, it doesn't mean the entire script is unused, it means a certain amount of the script's code isn't doing anything on the page.

The algorithm for finding unused code isn't perfect either, I often see scripts that aren't used at all on a page having some sort amount of used code according to google.

The "Remove Unused Javascript" should really be taken with a pinch of salt, and in cases of plugins, like youtube's API, it should be ignored since you can't edit a external API to be more efficient.

In your case, the file is very much needed if your using an embedded Youtube video.
If you want to optmize how you load your Youtube video, consider lazyloading it.