Thanks for the detailed write-up, Sarah!

One important clarification on just how alpha it is - currently the only things that get saved are menu names and the automatically add new items setting - and those both require a pending core patch, unfortunately. The screen options are in place and should work, though, it's in the four-square icon and they sync with the existing Menus screen. Much of the initial UI is in place and you can get a good sense of how everything might look once the underlying functionality is completed. I just updated the plugin description to reflect this as well.

One of the most important pieces to enabling the Menu Customizer to feel more manageable, and to improving the Widget Customizer experience at the same time, is a core Customizer "Pages" feature that groups several control sections together. We're working on that here: https://core.trac.wordpress.org/ticket/27406. Currently, we have core patches with a couple of different approaches and are awaiting feedback from lead devs on the API side and designers on the UI/UX side. I made a quick demo video too: https://cloudup.com/cpFF42uiW2l

While the ability to set theme locations has been in the customizer from the start, complete menu management in the customizer could be a gateway to exploring broader applications for the customizer's use, since menus are not as theme-centric as the current interface presents them. One idea I had recently was to merge all of WordPress core's settings pages into the customizer, utilizing the customizer pages concept mentioned above.

There's a lot of other customizer-related work going on in Core Trac right now (largely by Weston Ruter, who created the Widget Customizer), which should come together to really expand the possibilities here.

I'm really excited to have the opportunity to work on this project. Things are getting improved on a daily basis and hopefully it'll get to the point that it's feature-complete and ready for full-scale testing and feedback soon!