Using a framework like Backbone I currently have a page http://proctors.org/tv that is driven by several views and attachments with tons of jQuery and hacked-together JavaScript spaghetti, and I would like to clean it up. I understand that newer Javascript "MVC" frameworks make use of templates, and don't strictly store all the data in the DOM as you might do with jQuery.

I believe you would use something like Services and/or Views as the backend, with Backbone on the front.

There is a D.o project for Backbone, and I remember there being a BoF or Session at DrupalCon Denver about it...

I am hoping to discover if it is worth my time to try and rewrite this in a new framework or just clean up what I have here.
I have used both Ember and Backbone along with Drupal - but not in the way you would think.
Ember and Backbone are both fine tools to create a web application - but Drupal is not really a good tool to be used along with them. What you desire when you built a web app is something light and fast. You want it to be fast to be able to give your users the feeling if immediate  response times. You want it to be light to make it scale better. Drupal is not fast - the full Drupal bootstrap takes a long time (compared to a lot of other options) if all you want to do is provide a REST API for your web app. Drupal is PHP, which also makes it quite RAM hungry limiting the number of concurrent users.
So if Drupal is so bad for this why use it?
You should really use Drupal for what Drupal excels at - a CMS. What I've done in the past with projects that needed a web app is to use Drupal to create the CMS around the web app. To build stuff like blogs, content lists and the likes. I also created modules to define some backend stuff - like schema definitions, and other stuff where it made sense to use Drupal - because it made my development much easier.
Instead of using Drupal to create the REST API that the webapp needed - I used Node.js to create the actual REST API and other checks AJAX calls that the webapp needed to make. Node.js excels in doing just this. It's blazing fast (response times as low as 30ms for a list of objects). It's also very lightweight, since JavaScript on the server uses asynchronous actions, a Node.js server can usually handle thousands of concurrent users where PHP can handle maybe 100. In addition to that - since Node.js is JavaScript, you can reuse a lot of the code server and client side. You can literally write the same validation code, instead of having to implement both client side in JavaScript and server side in PHP. If you are going to Drupal con in Münich in a few weeks - you should consider watching the node.js presentation.
So if you are a developer, I would strongly recommend doing a similar setup, and only use Drupal for what it is really good at. If you want to make a simple webapp and you don't expect a lot of user, using Drupal as a backend could be an option if it helps your development time.

Update Drupal 8
So with the release of Drupal 8 some of the above still holds true but not in the same extend as earlier. Drupal 8 is like Drupal 7 not a light weight tool, like Node.js, Rails, Django etc. Since JavaScript apps mainly consume various services, you still should consider if Drupal is the best tool for this.
With Drupal 8 a lot of things have been improved. Services is a lot more native, with Symphony driving the request/response. Drupal 8 has a lot of interesting possibilities with with the advanced caching and all of the great features. But even with all of the greatness of Drupal 8, it still is a lot heavier than light weight applications. It's hard to tell how successful Drupal 8 can be used as a fast webservice provider.
So still, with all being said and done, my general recommendation stays the same. Being a Drupal developer, it's easy to use Drupal for everything, since we're familiar with it. It feels safe and Drupal can be used for just about anything. Remember while it's possible to use Drupal for anything, it doesn't mean that it's the best tool for everything. Doing a sound analysis and understanding the strengths and weaknesses of the tools you want to use on an IT project, will always be a great help and can help you avoid ending in a deadlocked position with a result that doesn't live up to your expectations.