Thanks, Sarah!
This would only be possible with the new Panels API
A correction: the plugin actually doesn't use the Panels API. It just registers one "Posts" section and adds controls to it. Using panels may be a better approach when editing multiple posts concurrently.
The plugin also grants customize capability to authors and editors who wouldn’t normally have access.
This new customize capability was introduced in 4.0 (#28605).
On themes like Twenty Fourteen, where you have featured images that appear on single posts, the plugin adds a convenient way to live preview various images without having to go back and forth between the admin and the frontend.
Actually, and I only realized this relatively recently myself, but did you know that changing the featured image actually makes the change live even before you save the post? I found this very surprising behavior. The only way to truly preview a change to a featured image is to use something like Customize Posts.

Likewise, something else I learned recently, did  you know that changing a page template cannot be previewed from the post editor? When you change the selected page template, the change will not show up when you preview the post (see #11049). However, in Customize Posts you can preview changes to the page template just by changing the dropdown selection, and then you can see what your page would look like with the new template after the preview refreshes.

Most other changes to metaboxes containing data that gets saved to custom fields (postmeta) also get written when clicking the Preview button. The Customize Posts plugin provides a way to get around this, and also provides a live preview of the changes. Fixing this underlying issue of incorrectly persisting postmeta when doing a preview is captured in #20299. The current Customize Posts UI for editing postmeta is just a non-user-firendly first stab; future iterations should make it easy to add new controls to manage postmeta, just as metaboxes are added to the post editor.
If a frontend editor does wind up landing in core in the future, do you think that the concept shown here for editing posts and postmeta could be a compliment to the experience?
I'm collaborating with @avryl on integrating the Front-end editor with the Customizer. We have a couple issues open where we've been discussing ways to collaborate, along with @celloexpressions. She's actually re-writing the Front-end editor to use the Customizer as the underlying framework for managing changes as well. So it's not an either/or but a both/and.

Customize all the things!