Hello John-Robert,
Embedded timelines cannot display your Home timeline because the Home timeline is private to you as a logged-in user. It includes protected accounts that cannot be publicised, and is subject to the filter settings you apply to those accounts (such as hiding retweets from select people.) A user viewing an embedded timeline would not be guaranteed to see the same content as you see in your home timeline. For this reason, we don’t offer a ‘Home timeline’ embed; it would cause confusion at its inconsistent edges.
Instead, for this purpose, you can use a list widget. You can curate the public accounts you want to be visible on your site, and the timeline will display them.
Ben