Your fiddle code does not show the issue, because it does not add click event handlers to anything except buttons. These are not announced as "clickable".
I just tried a bare bones case (sans Backbone). Yes, indeed, Firefox appends "clickable" to the accessible name of 'non-operable' elements with click handlers. It does not do this with operable elements such as buttons.
And this is not wrong, necessarily. The real question is: Why do you need click handlers on headings, tables and paragraphs anyway?
If you really need a heading to behave (semantically) like a heading, it should not have click event handlers. Headings have a very important role. Being clickable is not it. ("A role is a promise").
You can put a button inside a heading, and attach the click events to that. Being a button, it will be announced as "button", instead of as "clickable". Another alternative is to put a hyperlink inside the heading. This would be announced as a "link" of course. Both of these approaches provide reliable cues about how the element may be expected to behave. If clicking the heading toggles the visibility of the content it is heading for, then there is a formal pattern called "disclosure", provided by the w3c here which you should be using.
"Clickable" gives no such cues. But I guess you don't need clickable headings and paragraphs at all.
Adding click event handlers to every single 'View', even non-operable ones, just seems wrong to me. Perhaps you could just use the default View behavior for non-operable content, and reserve 'OperableView' for stuff that actually needs to be clickable.
If you can make this distinction (operable/non-operable), you'll have a far easier time implementing accessibility, because this distinction is at the heart of ARIA.
And your bogus 'clickable' announcements will automagically disappear.
