If you’re looking for a native WordPress solution I’d recommend using the Theme Customize API. It can be a bit overwhelming at first, but it’s worth the investment in learning how to use it. Once you understand the concepts and have built your own Theme Options, you’ll find it solves a lot of problems for you. In general, you could give the user a dropdown option for “Default Layout”, “Large Story”, and “Slow Featured”, etc.
Using get_template_part is a great way to toggle your layout.
As for allowing the user to “pick” the Large Story (or “picking” the 3-4 featured stories), I’d recommend creating a Custom Taxonomy (similar to a tag) so the user can “tag” a story as a “Large Story” or “Slow Featured Story”.
Then, in your “Large Story” template part, you can use WP_Query to fetch “posts with a term large-story, reverse chronological (newest), limit 1″…
And in your “Slow Day” template part you can use WP_Query to fetch “posts with a term slow-featured, reverse chronological (newest), limit 4″… etc.