How would you accomplish the following advanced settings page?

Based on what you want, if it were me, I would enable featured images and excerpts on pages and then build a widget to feature a page. Then put a sidebar on your homepage and use those widgets to reorder and select which pages to show in which order.

Then you’re managing all content related to a page on that page and your reordering experience is built for you via the widget interface. This is almost all using core features, so it would be pretty quick to implement.

Alternately, you could skip the widgets and use the menu_order and a custom taxonomy to select and order the pages to feature. The homepage would just use a custom query to grab and order the right pages while displaying the title, image, excerpt, and link.