You can't get to every single piece of functionality, but there's this -- https://archive.org/help/json.php -- and this -- https://archive.org/advancedsearch.php, which you can output as JSON/JSONP.

For instance, this broken piece of shit was my attempt to do something about the LMA interface -- http://www.archive-ui.org/#/. (select a show, reload the page, then it'll work. got busy and lost interest).

The interface is as bad as you say, but at least they give us the ability to do something about it ourselves.