"I want to write a bunch of gross code to perform dirty checking of objects to keep everything in sync" said no UI developer ever.
That's the point of libraries and frameworks. They do the repetitive or "ugly" or "hard" things that we don't want to do over and over.
Except that this is in no way an equivalent comparison. With Backbone, you actually have to write the hundreds of callbacks in your application code.
Whereas with Angular, you attach a value to a scope, and declare it in a template. In that case, the framework actually does take care of the 'ugly' for you.
Backbone keeps the ugly in front of your face, it just gives you a slightly neater way to organize it (i.e. as compared to jQuery soup).